Question

date: per:
classify each statement as true or false. if false, write the correct statement.

1. in a 45 - 45 - 90 triangle, if a leg has a length of x the hypotenuse is 2x.
2. the hypotenuse of a 30 - 60 - 90 triangle is twice as long as the shorter leg.
3. in a 45 - 45 - 90 triangle is also known as an isosceles right triangle.
4. in a 30 - 60 - 90 triangle, the longer leg is \\(\sqrt{3}\\) times the shorter leg.
5. \\(x = \\)

\\(y = \\)

1. \\(x = \\)

\\(y = \\)

1. \\(x = \\)

\\(y = \\)

1. \\(x = \\)

\\(y = \\)

1. \\(x = \\)

\\(y = \\)

1. \\(x = \\)

\\(y = \\)
\\(z = \\)

1. what is the length of the diagonal of this square? give your answer in exact simplified radical form.

Explanation:

Step1: Evaluate 45-45-90 hypotenuse rule

In a 45-45-90 triangle, hypotenuse = $x\sqrt{2}$ (not $2x$). So statement 1 is FALSE. Correct: Hypotenuse is $x\sqrt{2}$.

Step2: Evaluate 30-60-90 hypotenuse rule

In a 30-60-90 triangle, hypotenuse = 2×shorter leg. Statement 2 is TRUE.

Step3: Evaluate 45-45-90 triangle definition

A 45-45-90 triangle has two equal legs, so it is an isosceles right triangle. Statement 3 is TRUE.

Step4: Evaluate 30-60-90 longer leg rule

In a 30-60-90 triangle, longer leg = $\sqrt{3}×$shorter leg. Statement 4 is TRUE.

Step5: Solve 45-45-90 triangle (Q5)

Legs = 10, so $x=10$. Hypotenuse $y=10\sqrt{2}$.

Step6: Solve 45-45-90 triangle (Q6)

Hypotenuse =7, so leg $x=\frac{7}{\sqrt{2}}=\frac{7\sqrt{2}}{2}$. $y=7$ (other leg matches angle 45°).

Step7: Solve 30-60-90 triangle (Q7)

Shorter leg=16, longer leg $y=16\sqrt{3}$. Hypotenuse $x=32$.

Step8: Solve 45-45-90 triangle (Q8)

Legs = $x$, hypotenuse=6. So $x\sqrt{2}=6 \implies x=3\sqrt{2}$. $y=3\sqrt{2}$.

Step9: Solve 30-60-90 triangle (Q9)

Shorter leg=11, hypotenuse $y=22$. Longer leg $x=11\sqrt{3}$.

Step10: Solve split triangle (Q10)

Left 45-45-90: leg=24, hypotenuse $x=24\sqrt{2}$. Right 30-60-90: shorter leg=12, longer leg $y=12\sqrt{3}=4\sqrt{3}×3$? Correction: Split base=14, left segment=12, right=2. 30-60-90 shorter leg=2, longer leg $y=2\sqrt{3}$? No, correct: Height=12. 45-45-90: $x=12\sqrt{2}$. 30-60-90: shorter leg=12, hypotenuse $y=24$, longer leg $12\sqrt{3}$.

Step11: Square diagonal calculation (Q11)

Square side=6, diagonal = $6\sqrt{2}$.

Answer:

1. FALSE; Correct statement: In a 45-45-90 triangle, if a leg has length $x$, the hypotenuse is $x\sqrt{2}$.
2. TRUE
3. TRUE
4. TRUE
5. $x=10$, $y=10\sqrt{2}$
6. $x=\frac{7\sqrt{2}}{2}$, $y=7$
7. $x=32$, $y=16\sqrt{3}$
8. $x=3\sqrt{2}$, $y=3\sqrt{2}$
9. $x=11\sqrt{3}$, $y=22$
10. $x=12\sqrt{2}$, $y=12\sqrt{3}$
11. $6\sqrt{2}$ cm