Question 328668: The altitude to the hypotenuse of a right triangle divides the hypotenuse into two
segments. The length of one segment exceeds the length of the other segment by 15. If the length of the altitude is 10, find the area of the largest triangle.
Answer by J2R2R(94)   (Show Source):
You can put this solution on YOUR website! The original triangle ABC where A is the right angle and D is the point on the hypotenuse (BC) where the altitude drops to.
Sketch this and you will see we have one large right angled triangle (ABC) comprising of two smaller right angled triangles (ABD and ACD). Let BD = x and from the information given CD = x + 15.
Now the hypotenuses of the smaller triangles (AB and AC) are the sides making the right angle of the large triangle so we can apply Pythagoras twofold. Firstly to get the hypotenuse of both smaller triangles then using these to find the hypotenuse of the large triangle.
AB^2 = x^2 + 100 (Pythagoras)
AC^2 = (x + 15)^2 + 100 = x^2 + 30x + 325 (Pythagoras)
Now combining these hypotenuses as smaller sides in the bigger triangle
BC^2 = (2x + 15)^2 = AB^2 + AC^2 (Pythagoras)
4x^2 + 60x + 225 = x^2 + 100 + x^2 + 30x + 325
4x^2 + 60x + 225 = 2x^2 + 30x + 425
2x^2 + 30x - 200 = 0
X^2 + 15x - 100 = 0
(x + 20)(x - 5) = 0
Solutions of this are x = 5 or -20.
Now lengths are positive values so we will use x = 5 to give BD = 5 and CD = 20 which gives BC = 25.
The sides of the triangles ABD, ACD, ABC are listed below. Think of the square of the hypotenuse being the sum of the squares of the other two sides:
5, 10, root125;
20, 10 root500;
root125, root500, root625 (= 25)
5 and 20 makes 25.
You requested the area of the largest triangle - I take it you mean the larger of the two smaller triangles being the one with sides 10 and 20 at right angles. Area = 0.5 times length times height = 100 square units.
The largest triangle overall is the one comprising both smaller ones, so the area of this triangle with sides root125 and root500 at right angles is 0.5 root125 root500 = 0.5 root(5 cubed times 5 times 10 squared) = 0.5 x 250 = 125 square units. I don’t think you meant this one but when two triangles are combine to make a bigger triangle then this is the biggest triangle.
NB we could have arrived at the same answers taking the negative value -20 giving BD = -20 and CD = -5 so we still get a hypotenuse of -25 but these are all negatives of the originals. So we don’t gain anything with negative lengths since we use lengths as positive.
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