Question

In Δ A B C, the measure of the largest angle is 32 less than 3 times the smallest angle. The measure of the middle angle is 8 more than half the measure of the largest angle. What is the measure of the middle angle?

Answer 1

Answer:

52degrees

Step-by-step explanation:

Let the measure of the largest angle be x

middle angle be y

smallest angle be z

Since the sum of angle in a triangle is 180 degrees, hence;

x + y + z = 180 .... 1

If the measure of the largest angle is 32 less than 3 times the smallest angle, then;

x = 3z - 32

3z = x+ 32

z = x+32/3 ..... 2

If he measure of the middle angle is 8 more than half the measure of the largest angle, then;

y = x/2 + 8 .... 3

Substitute 2 and 3 into 1;

x + x/2 + 8 + (x+32)/3 = 180

multiply through by 6

6x + 3x + 48 + 2(x+32) = 1080

9x + 48 + 2x + 64 = 1080

11x + 112 = 1080

11x = 1080-112

11x = 968

x = 968/11

x = 88 degrees

Get the middle angle y;

Since y = x/2 + 8

y = 88/2 + 8

y = 44 + 8

y = 52 degrees

Hence the measure of the middle angle is 52degrees

Answer 2

The angles in a triangle add up to 180 degrees.

The middle angle is 52 degrees

Let the largest, middle angle and the smallest angle be x, y and z

From the question, we have:

$\mathbf{x = 3z -32}$

$\mathbf{y = 8 + \frac 12x}$

Make z in $\mathbf{x = 3z -32}$

$\mathbf{z =\frac{x + 32}{3}}$

The angles in a triangle add up to 180 degrees.

So, we have:

$\mathbf{x + y + z = 180}$

Substitute the expressions for y and z in   $\mathbf{x + y + z = 180}$

$\mathbf{x + 8 + \frac 12x + \frac{x+32}{3} = 180}$

Multiply through by 6

$\mathbf{6x + 3x + 48 + 2(x+32) = 1080}$

$\mathbf{9x + 48 + 2(x+32) = 1080}$

Open brackets  

$\mathbf{9x + 48 + 2x + 64 = 1080}$

Evaluate like terms

$\mathbf{11x + 112 = 1080}$

Subtract 112 from both sides

$\mathbf{11x = 968}$

Divide both sides by 11

$\mathbf{x = 88}$

Substitute 88 for x in $\mathbf{y = 8 + \frac 12x}$

$\mathbf{y = 8 + \frac{88}2}$

$\mathbf{y = 8 + 44}$

$\mathbf{y = 52^o}$

Hence, the middle angle is 52 degrees

Read more about triangles at:

https://brainly.com/question/22790855