The legs of a right triangle are 8 and 15. Find the altitude to the hypotenuse.

Guest Jun 10, 2020

#1**+1 **

Hello! I'm working on the problem now. There's quite a bit of steps. However, it is possible. Don't get discouraged. First you need to determine the length of the hypotenuse using the Pythagorean theorem. It's length is 17.

From there, I looked up how to do altitude and you need the height of the triangle so I decided to use this formula:

a = 1/2 * bh

Before using this formula, you need to calculate the area first. For this, I used Heron's Formula. You need to find the semi perimeter, so to get this, add all of the sides together & divide by 2. You should get 20. Then I got the area by using Heron's Formula:

\(\sqrt{20(20-17)(20-15)(20-8)} = \sqrt{20(3)(5)(12)} = \sqrt{3600} = 60\)

Now that you have your area, 60, you can use the formula above, a = 1/2 * bh. Just substitute your area into the equation and solve for you height.

\(60 = \frac{1}{2}(14)h\)

\(60 = 7h\)

\(h = 8.57\)

(Will continue... still solving)

auxiarc Jun 11, 2020

#2**0 **

I'll attempt this problem in a different way ...

Draw right triangle(ABC) with C the right angle.

Let AC = 8 and BC = 15.

Using the Pythagorean Theorem AB = 17.

Drop a perpendicular from angle(C) to the hypotenuse AB.

Label the point of intersection P.

PC is the height.

By similar triangles (or a theorem from your textbook):

PA / AC = AC / AB ---> PA / 8 = 8 / 17 ---> PA = 64 / 17

PB / BC = BC / BA ---> PB /15 = 15 / 17 ---> PB = 225 / 17

Continuing with similar trianges (or another theorem):

AP / PC = PC / PB ---> ( 64 / 17 ) / PC = PC / ( 225 / 17 )

cross-multiplying: ( 64 / 17 ) · ( 225 / 17 ) = PC^{2}

( 64 · 225 ) / ( 17 · 17 ) = PC^{2}

14,400 / 289 = PC^{2}

120 / 17 = PC

as a decimal: 7.0588...

geno3141 Jun 11, 2020

#3**+1 **

**The legs of a right triangle are 8 and 15. Find the altitude to the hypotenuse.**

Formula: \( h_c = \dfrac{ab}{c}\) (right triangle)

\(\begin{array}{|rcll|} \hline \mathbf{h_c} &=& \mathbf{\dfrac{ab}{c}} \quad | \quad c = \sqrt{a^2+b^2} \\\\ h_c &=& \dfrac{ab}{\sqrt{a^2+b^2}} \\\\ h_c &=& \dfrac{8*15}{\sqrt{8^2+15^2}} \\\\ h_c &=& \dfrac{120}{\sqrt{289}} \\\\ \mathbf{h_c} &=& \mathbf{\dfrac{120}{17}} \\ \hline \end{array}\)

heureka Jun 11, 2020

#4**0 **

Have you heard of the "inverse Pythagorean theorem"?

Let a and b be the legs of a right triangle, and h be the altitude to the hypotenuse.

The inverse Pythagorean theorem states that \(h^{-2} = a^{-2} + b^{-2}\).

We can use this equation to solve this problem.

Let h be the altitude to the hypotenuse.

\(\dfrac1{h^2} = \dfrac1{8^2} + \dfrac1{15^2}\\ h^2 = \dfrac{120^2}{8^2 + 15^2} = \left(\dfrac{120}{17}\right)^2\\ h = \dfrac{120}{17}\)

MaxWong Jun 11, 2020