Questions on Geometry: Triangles answered by real tutors!

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Question 1164146: Two sides of a triangle are AB=34cm and AC=25cm and their included angle measure 62°. Question is, Find the distance of the orthocenter to side AB.
Click here to see answer by ikleyn(52747)

Question 1178105: The longest side of a triangular building lot has length 190 meters and the next longer side is 180 meters (the shortest length is unknown). The angle between the longer sides measures 54º. Using the median to the longest side (the segment joining the midpoint of the side to the opposite vertex) the lot is divided into two lots with equal area. Find the length of the median to the nearest tenth of a meter.
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Question 1210262: Let ABC be a triangle with side lengths AB = 5, BC = 6, and AC = 9. What is the area of the triangle with side lengths \tan A$, \tan B, and \tan C?

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Question 1210260: Find the area of the triangle with side lengths 4, 5, and 8.

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Question 1210242: In the diagram below, \overline{AD} and \overline{BE} are angle bisectors of \angle BAC and \angle ABC, respectively, and they intersect at T. We know that BD=12, AE=8, and BF=3+AE. Find AB.

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Question 1210243: In the diagram, \overline{AD} is an altitude, \overline{BE} is a median, and \overline{AD}, \overline{BE}, and \overline{CF} are concurrent. If AE = 11, BE = 9, and AD = 6\sqrt{2}, what is AF?

Click here to see answer by CPhill(1959)

Question 1210193: Triangle ABC has AB = 6. Let D lie on BC such that \overline{AD} bisects \angle BAC. If BD = 3 and CD = 5, what is CD?
Click here to see answer by greenestamps(13195)

Question 1210193: Triangle ABC has AB = 6. Let D lie on BC such that \overline{AD} bisects \angle BAC. If BD = 3 and CD = 5, what is CD?
Click here to see answer by ikleyn(52747)

Question 1210193: Triangle ABC has AB = 6. Let D lie on BC such that \overline{AD} bisects \angle BAC. If BD = 3 and CD = 5, what is CD?
Click here to see answer by CPhill(1959)

Question 1210190: Chris places an orange cone at his current location. Then, he faces west, walks 40 meters, turns 30^{\circ} to his right, and walks 20 meters. How far is Chris from the cone, in meters? Round your answer to the nearest whole number.
(You will need to use a calculator.)

Click here to see answer by CPhill(1959)

Question 1210147: In \triangle ABC, we have AB = AC = 4 and \angle BAC = 45^\circ. If M is the midpoint of BC, then find AM^2.
Click here to see answer by ikleyn(52747)

Question 1210147: In \triangle ABC, we have AB = AC = 4 and \angle BAC = 45^\circ. If M is the midpoint of BC, then find AM^2.
Click here to see answer by CPhill(1959)

Question 1210137: The left column contains pairs of triangles with different side-length and angle-measure congruences marked. Match each diagram in the left column with a congruence/similarity criterion in the right column that justifies why the two triangles are congruent/similar. Duplicates are NOT permitted. Note: to undo a connection, click the link itself, not one of the boxes in the columns.
Click here to see answer by CPhill(1959)

Question 1210138: The left column contains pairs of triangles with different side-length and angle-measure congruences marked. Match each diagram in the left column with a congruence/similarity criterion in the right column that justifies why the two triangles are congruent/similar. [b]Duplicate answers ARE permitted[/b]. (Note: to undo a connection, click the link itself, not one of the boxes in the columns.)
Click here to see answer by CPhill(1959)

Question 1210139: In the diagram, \overline{AD} is an altitude, \overline{BE} is a median, and \overline{AD}, \overline{BE}, and \overline{CF} are concurrent. If AB = 11, AC = 12, and AD = 6, what is AF?

Click here to see answer by CPhill(1959)

Question 1210136: In the diagram below, we know \tan \theta = \frac{3}{4}. Find the area of the triangle.

The triangle is right, and the hypotenuse is 80.
Click here to see answer by math_tutor2020(3816)

Question 1169820: Give a formal proof of the following theorem
1. A quadrilateral with one pair of opposite sides equal and parallel is a parallelogram. (Construct a diagonal).
2. If both pairs of opposite sides of a quadrilateral are equal, the quadrilateral is a parallelogram. (Construct a diagonal).
3. The diagonals of a rhombus, (i) bisect each other at right angles, (ii) bisect the angles of the rhombus.
Click here to see answer by CPhill(1959)

Question 1169822: Please help me to solve this.. DEF is a triangle with angle EDF=2x°. Line DE and line DF are produced to G and H respectively so that side EF=side EG=side FH. Line EH and line FG intersect at K. Show that angle EKG=90-x°. Thank you
Click here to see answer by CPhill(1959)

Question 1168324: In a basketball league of x teams of which every team plays every other twice, the total number of games played is x²-x
a. How many teams are there in a league the plays a total of 72 games?
b. If there were 6 teams in the league, how many in all would be played?
Click here to see answer by ikleyn(52747)

Question 1174508: Astronomers often measure large distances using astronomical units (AU) where 1 AU is the average distance from Earth to the Sun. In the image, d represents the distance from a star to the Sun. Using a technique called “stellar parallax,” astronomers determined \large \theta is 0.00001389 degrees. How far away is the star from the Sun in astronomical units? Show your reasoning.
Click here to see answer by CPhill(1959)

Question 1174947: Show that if ∆A'B'C' is the image of ∆ABC under a dilation with center O and scale
factor k, then
∆A'B'C' ∆ABC = k
2
Click here to see answer by CPhill(1959)

Question 1209580: Could someone outline a strategy for solving the following problem?
*Let $ \triangle ABC $ be an acute-angled triangle with circumcenter $ O $, incenter $ I $, and nine-point center $ N $. Let the incircle of $ \triangle ABC $ touch the sides $ BC $, $ CA $, and $ AB $ at $ D $, $ E $, and $ F $ respectively, and let $ A' $, $ B' $, $ C' $ be the midpoints of the arcs $ BC $, $ CA $, and $ AB $ (not containing the opposite vertices) on the circumcircle. Define points $ P $, $ Q $, and $ R $ as follows:*
- *Draw the line through $ I $ parallel to $ BC $; let it meet the circumcircle (other than $ A $) at $ P $.*
- *Similarly, let the line through $ I $ parallel to $ CA $ meet the circumcircle (other than $ B $) at $ Q $, and the line through $ I $ parallel to $ AB $ meet the circumcircle (other than $ C $) at $ R $.*
*Now, let $ X $ be the intersection of lines $ A'P $ and $ B'Q $, and let $ Y $ be the intersection of lines $ B'Q $ and $ C'R $. Suppose further that:*
1. *The circle $ \omega_1 $ through $ D $, $ E $, $ F $ (the incircle contact points) is tangent to the circle $ \omega_2 $ through $ I $, $ X $, and $ Y $.*
2. *The line through $ I $ perpendicular to $ XY $ meets side $ BC $ at $ T $.*
*Prove that:*
- *(a) $ T $ is the midpoint of $ BC $, and*
- *(b) The radical axis of the incircle and the circumcircle of $ \triangle ABC $ is parallel to the Euler line of $ \triangle ABC $.*
What would be an effective approach or roadmap to untangle and eventually prove these assertions?
Click here to see answer by CPhill(1959)

Question 1209528: (58) Triangle ABC has median AD. If the area of triangle ABD is 18, then what is the area of triangle ADC?
Link to Diagram: https://ibb.co/FCJYq1K
Click here to see answer by ikleyn(52747)

Question 1209530: Triangle ABC is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are indicated. Find the area of triangle ABC.
Link to diagram: https://ibb.co/Kc4QKqdW
Click here to see answer by ikleyn(52747)

Question 1209531: (64) Triangle ABC has AD ⊥ BC and BE ⊥ AC. If AD = 20, BC = 29 and BE = 21, then find the length of AC.
Link to diagram: https://ibb.co/zc6R9gz
Click here to see answer by ikleyn(52747)

Question 1191559: Let's say, a man has to check the schedule of the boat trips at the information center, A. The 200-m path to the information center and the 400-m path to the boat rental dock, B, intersect at the parking lot, C, forming a right angle. He walks straight from the parking lot to the lake D as shown, where a sign tells him that he is approximately 357.77 m from the dock. How far is the man from the information center? What is the equation to represent the situation? If the man will go back to the parking lot from the information center. What is total distance he walked?
Click here to see answer by ikleyn(52747)

Question 1209517: (32) If QT:TU = 4:1 and UW:WP = 3:1, what fraction is triangle PWT of triangle PQU?
Diagram: https://ibb.co/9kP53CbZ
Click here to see answer by ikleyn(52747)

Question 1209522: (44) Find the altitude of the triangle to side AB.
Link to diagram: https://ibb.co/spznGBS7
Click here to see answer by ikleyn(52747)

Question 1209435: Equilateral triangle ABC with sides of 1cm has altitude AD. median AE of triangle ABD is drawn. What is the area, in cm^2, of triangle AEC?
Click here to see answer by math_tutor2020(3816)

Question 1209423: the smallest angle of a triangle with sides 9cm, 11cm and 13 cm

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Question 1209390: The sides of a triangle measure 4, 9, and 11. Is this a right triangle? If not then is it acute or obtuse?
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Question 1209348: In triangle XYZ, X_1, Y_1, and Z_1 are the midpoints of $\overline{YZ},$ $\overline{XZ},$ and $\overline{XY},$ respectively. A dilation maps $X$ to $Y$ with scale factor 3. A second dilation maps $Y$ to $Z_1$ with scale factor 5. When both dilations are combined, what is the overall scale factor?

Click here to see answer by math_tutor2020(3816)

Question 1198213: While rounding the bases on a home run, a baseball player makes an x (point A) in the dirt 1/3 of the way from 1st to second base. Then he ran another 1/3 of the distance and made an x (point B) in the dirt. The points from A to B make a triangle with point H (home plate). What is the area and perimeter of the triangle HAB?
Click here to see answer by onyulee(41)

Question 1192682: Point D is the midpoint of median AM of triangle ABC. Point E is the midpoint of AB, and point T is the intersection of BD and ME. Find the area of triangle DMT if [ABC] = 180.
Click here to see answer by textot(100)

Question 1209140: In a word processing document or on a separate piece of paper, use the guide to construct
a two column proof proving SMRN, given RMSN and ∠MRS ∠NSR. Upload the entire proof below.
Given:
RMSN
∠MRS ∠NSR
Prove:
SMRN

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Question 1209091: Find the area of the following triangle, in cm2:
https://ibb.co/TWY0pdw
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Question 1208950: prove that the line drawn from the vertex of an isosceles triangle through the point of intersection of 2 medians drawn from the base angle is perpendicular to the base. Explain with statement s and reason
Click here to see answer by KMST(5328)

Question 1208911: Equilateral triangle ABC has sides of 14 cm. A circle of radius 2 cm inside it is tangent to sides AB & AC. Find the distance from the circle's centre to side BC, in cm.

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Question 1208906: The area of a triangle is divided into 3 equal parts by line segments parallel to one side. If the length of that side is 18 cm, find the length of the longest line segment inside the triangle, in cm.

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Question 1208907: In the diagram, AB is the diameter of the circle, and E is the centre. Find the measure of angle D, in degrees.
https://ibb.co/ZKnPnRB
Click here to see answer by ikleyn(52747)

Question 1208855: The ratio of the complements of two angles is 3:2, and the ratio of their supplements is 9:8. Find the two original angles.
Click here to see answer by greenestamps(13195)

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