Question

Im in 10th grade geometry and need help

Answer 1

Answer:

x = 15

$\displaystyle\mathsf{Hypotenuse\:(y)\:=\:15\sqrt{2}}$

Step-by-step explanation:

We are given a diagram of a 45°-45°-90° right triangle, for which one of its legs has a measure of 15.  The prompt also requires us to determine the values of the right triangle's other leg (x), and hypotenuse (y).

45°-45°-90° Triangle Theorem:

We can apply the 45°-45°-90° Triangle Theorem in this given problem, which states that in a 45°-45°-90° right triangle, the measure of its hypotenuse is $\displaystyle\mathsf{\sqrt{2}}$ times the length of its other two legs.  

Solution:

Solve for y (hypotenuse):

In reference to the 45°-45°-90° Triangle Theorem, we can substitute the value of its given side length in order to determine the value of its hypotenuse:

$\displaystyle\mathsf{Hypotenuse\:(y)\:=\:leg\:\times\sqrt{2}}$

$\displaystyle\mathsf{Hypotenuse\:(y)\:=\:15\:\times\sqrt{2}}$

$\displaystyle\mathsf{Hypotenuse\:(y)\:=\:15\sqrt{2}}$

Solve for x (missing leg):

We can use the previous method to find the value of x, by referencing the 45°-45°-90° Triangle Theorem.  The only difference is that we are solving for the value of the other leg, "x."

$\displaystyle\mathsf{Hypotenuse\:(y)\:=\:leg\:\times\sqrt{2}}$

$\displaystyle\mathsf{15\sqrt{2}\:=\:(x)\:\times\sqrt{2}}$

Next, divide both sides by $\displaystyle\mathsf{\sqrt{2}}$ to isolate x:

$\displaystyle\mathsf{\frac{15\sqrt{2}}{\sqrt{2}}\:=\:\frac{(x)\:\times\sqrt{2}}{\sqrt{2}}}$

15 = x

Our solution for the value of x proves that the given diagram is indeed a 45°-45°-90° right triangle, as both of its legs have the same length of 15, and its hypotenuse is $\displaystyle\mathsf{\sqrt{2}}$ times the measure of its legs.

Final Answer:

Therefore, the value of x = 15, and $\displaystyle\mathsf{y\:=\:15\sqrt{2}}$.

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Keywords:

Right triangles

45°-45°-90° Triangle

Isosceles right triangles

Special right triangles

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Learn more about this topic here:

brainly.com/question/3960357

Answer 2

Answer: x = 15 ; y = 15√2

Concept:

The given triangle in the question is a 45°-45°-90° triangle.

It is considered to be a special triangle where the ratio of side lengths is 1:1:√2, meaning that the legs that correspond with 45° are 1, while the hypotenuse which corresponds with 90° is √2.

Please refer to the attachment below for a graphical explanation.

Given:

A 45°-45°-90° triangle

Leg₁: 15

Solve:

Knowing that the ratio of the triangle is 1 : 1 : √2

Find the value of x

x leg correspond with 45° angle

Leg₁ correspond with 45° angle

Leg₁ = 15

∵ The given ratio is 1 : 1 for 45° angle

$\boxed{x=15}$

Find the value of y

y hypotenuse correspond with 90° angle

Leg₁ correspond with 45° angle

Leg₁ = 15

∵ The given ratio is 1 : √2 for 45° : 90° angle

$\boxed{y=15\sqrt{2} }$

Hope this helps!! :)

Please let me know if you have any questions