There are two noncongruent triangles that have a 9-inch side, a 10-inch side, and that enclose 36 square inches of area. Find the length of the third side in each of these triangles.

If the angle θ between the sides is acute, then

(9 sinθ)(10)/2 = 36
θ = 53.13°
and the third side s comes from
s^2 = 9^2+10^2-180cos53.13°
s = √73

The angle might be obtuse. A little thought should show how to handle that case.

iki

Why did the triangle go to the party alone? Because it couldn't find its congruent angles!

But to answer your question, let's call the length of the missing side in the first triangle "x." Using the triangle area formula, we can set up the equation: (9 * x * sinθ)/2 = 36, where θ is the angle between the sides of length 9 and x.

Now, let's calculate the length of the third side in the first triangle using this equation. Of course, if we want the second triangle, we'll have to repeat the process. But first, let's solve this equation and then I'll help you find the length of the third side in the second triangle too!

To find the length of the third side in each of the noncongruent triangles, we can use the formula for the area of a triangle.

The formula for the area of a triangle is:

Area = (1/2) * base * height

In this case, we are given the lengths of two sides of the triangle (9 inches and 10 inches), and the area (36 square inches). Let's call the length of the third side x.

We can set up two equations:

Equation 1: (1/2) * 9 * x = 36
Equation 2: (1/2) * 10 * x = 36

Simplifying each equation:

Equation 1: 4.5 * x = 36
Equation 2: 5 * x = 36

Now, let's solve for x in each equation:

Equation 1: 4.5 * x = 36
Divide both sides by 4.5:
x = 36 / 4.5
x ≈ 8

Equation 2: 5 * x = 36
Divide both sides by 5:
x = 36 / 5
x = 7.2

Therefore, the length of the third side in the first triangle is approximately 8 inches, and in the second triangle is 7.2 inches.

To find the length of the third side in each of these triangles, we can use the formula for the area of a triangle:

Area = (1/2) * base * height

Let's assume the length of the third side in the first triangle is x inches. Since we know two sides (9 inches and 10 inches) and the area (36 square inches), we can set up an equation:

36 = (1/2) * 9 * x

Simplifying the equation, we have:

36 = 4.5x

Dividing both sides by 4.5, we get:

x = 8

So, the length of the third side in the first triangle is 8 inches.

Now, for the second triangle, we want it to be noncongruent to the first triangle. Since we have the same sides and area, we need to find a different length for the third side. Let's assume the length of the third side in the second triangle is y inches.

Setting up the equation using the same formula, we have:

36 = (1/2) * 10 * y

Simplifying the equation, we have:

36 = 5y

Dividing both sides by 5, we get:

y = 7.2

So, the length of the third side in the second triangle is 7.2 inches.