Question

Triangle ABC is given where m(angle)A=33 degrees, a=15 in., and the height, h, is 9 in. How many distinct triangles can be made with the given measurements? Explain your answer.

Answer 1

Answer:

One triangle

Step-by-step explanation:

Find the measures of the triangle ABC

Let

D ----> intersection point of the height triangle ABC with the segment AB

step 1

Find the measure of angle B

In the right triangle BCD

$sin(B)=\frac{CD}{BC}$ ---> by SOH (opposite side divided by the hypotenuse)

substitute the given values

$sin(B)=\frac{9}{15}$

using a calculator

$B=sin^{-1}(\frac{9}{15})=36.9^o$

step 2

Find the length of segment DB in the right triangle BCD

Applying the Pythagorean Theorem

$BC^2=DC^2+DB^2$

substitute

$15^2=9^2+DB^2$

$DB^2=15^2-9^2$

$DB^2=144\\DB=12\ in$

step 3

Find the measure of angle C

we know that

The sum of the interior angles in any triangle must be equal to 180 degrees

so

$A+B+C=180^o$

substitute the given values

$33^o+36.9^o+C=180^o$

$C=180^o-69.9^o=110.1^o$

step 4

Find the length side AC

In the right triangle ACD

$sin(A)=\frac{CD}{AC}$ ---> by SOH (opposite side divided by the hypotenuse)

substitute the given values

$sin(33^o)=\frac{9}{AC}$

solve for AC

$AC=\frac{9}{sin(33^o)}=16.5\ in$

step 5

Find the length of segment AD

In the right triangle ACD

$tan(A)=\frac{CD}{AD}$ ---> by TOA (opposite side divided by the adjacent side)

substitute the given values

$tan(33^o)=\frac{9}{AD}$

$AD=\frac{9}{tan(33^o}}=13.9\ in$

step 6

Find the length side AB

$AB=AD+DB$

substitute the given values

$AB=13.9+12=25.9\ in$

step 7

The measures of triangle ABC are

$A=33^o\\B=36.9^o\\C=110.1^o\\AC=16.5\ in\\CB=15\ in\\AB=25.9\ in$

so

The measurements of the ABC triangle are unique

therefore

Only one triangle can be constructed  with the given measurements