SOLUTION: 1.the sides of a triangle measures 20cm, 25,cm and 30 cm. Find the length of the altitude to the shortest side. Find the length of the median to the longest side. find the lengt

Algebra ->  Triangles -> SOLUTION: 1.the sides of a triangle measures 20cm, 25,cm and 30 cm. Find the length of the altitude to the shortest side. Find the length of the median to the longest side. find the lengt      Log On


   

Question 927442: 1.the sides of a triangle measures 20cm, 25,cm and 30 cm.
Find the length of the altitude to the shortest side.
Find the length of the median to the longest side.
find the length of the angle bisector of the smallest angle.
find the length of the line segment joining the midpoints of the 20-cm and 30-cm sides.

THANK YOUU.
Answer by MathLover1(20850) About Me  (Show Source):
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1.the sides of a triangle measuresa=20cm,+c=25cm, and b=30cm.

find the length of the angle bisector of the smallest angle.
find the length of the line segment joining the midpoints of the 20-cm and 30-cm sides.


Find the length of the altitude to the shortest side:
You know the lengths of 3 sides.
a=20cm,+c=25cm, and b=30cm
h is the altitude
Use Heron's formula to find out the area
A+=+sqrt%28+s+%28+s-a%29+%28+s-b%29+%28+s-c%29%29 ........where+s+= semi-perimeter and a,b, and c are side lengths
s+=+%2820cm%2B25cm%2B30cm%29%2F2=75cm%2F2=37.5cm
A+=+sqrt%2837.5+%28+37.5-20%29+%2837.5-25%29+%2837.5-30%29%29
A+=+sqrt%2837.5+%28+17.5%29+%2812.5%29+%287.5%29%29
A+=+sqrt%2861523.4375%29
A+=+248.04 then use
248.04+=+%281%2F2%29+%2A+20%2A+h
248.04+=+10%2A+h
h+=+24.8+cm ANSWER
Find the length of the median to the longest side:
the longest side is b=30+cm
Draw triangle ABC, with
AB+=+25cm, BC+=+20cm, AC+=+30cm
Using law of cosines, we get
c%5E2+=+a%5E2+%2B+b%5E2-2%2Aa%2Ab%2Acos%28C%29
25%5E2+=+20%5E2+%2B+30%5E2-2%2A20%2A30%2Acos%28C%29
625=+400%2B+900-1200%2Acos%28C%29
625=+1300-1200%2Acos%28C%29
1200%2Acos%28C%29=+1300-625
1200%2Acos%28C%29=+675
cos%28C%29=+675%2F1200
cos%28C%29=+0.5625

Draw median from B to AC at point M.
Now we have triangle BCM with BC+=+20cm, CM+=+15cm
We will find BM using law of cosines:
BM%5E2=+BC%5E2+%2B+CM%5E2+-+2+%28BC%29%28+CM%29+cos%28C%29
BM%5E2=+20%5E2+%2B+15%5E2-+2+%2820%29%2815%29+%280.5625%29
BM%5E2=+400%2B+225+-+337.5

BM%5E2=+287.5
Median BM=+sqrt%28287.5%29=> BM=16.96cm ANSWER


find the length of the angle bisector of the smallest angle.
Then the length of the angle bisector d is given by:
d%5E2=%28ac%2F%28a%2Bc%29%5E2%29%28%28a%2Bc%29%5E2-b%5E2%29+

d%5E2=%2820%2A25%2F%2820%2B25%29%5E2%29%28%2820%2B25%29%5E2-30%5E2%29+
d%5E2=%28500%2F%2845%29%5E2%29%28%2845%29%5E2-30%5E2%29+
d%5E2=%28500%2F%282025%29%29%282025-900%29+
d%5E2=0.25%281125%29+
d%5E2=281.25
d=16.77 ANSWER

find the length of the line segment joining the midpoints of the 20-cm and 30-cm sides.
The mid-segment of a triangle (also called a midline) is a segment joining the midpoints of two sides of a triangle.
Properties:
1.
The mid-segment of a triangle joins the midpoints of two sides of a triangle such that it is parallel to the third side of the triangle.

2.
The mid-segment of a triangle joins the midpoints of two sides of a triangle such that its length is half the length of the third side of the triangle.
so, the third side is 25cm and the half of the third side is 12.5cm ANSWER