15. what is the length of the hypotenuse of a right triangle with legs measuring 7 units and 24 units?
a. 25 units
b. 26 units
c. 27 units
d. 25 units
16. if triangle abc is similar to triangle def and the length of ab is 5 units while the length of de is 10 units, what is the scale factor from triangle abc to triangle def?
a. 10
b.
c. 2
d. 4
17. if a triangle is divided into two smaller triangles by an altitude and all three triangles are similar, what theorem could this be used to prove?
a. sss criterion
b. pythagorean theorem
c. hl criterion
d. sas criterion
18. rectangle r with side lengths 4 and 6 is dilated by a scale factor of 2. what are the side lengths of the resulting rectangle?
a. 2 and 3
b. 6 and 9
c. 8 and 12
d. 10 and 15
19. a ladder leaning against a building creates a right triangle with the ground. the ladder is 20 ft long, and the base of the ladder is 16 ft from the building. how high does the ladder reach on the building?
a. 10 ft
b. 15 ft
c. 11 ft
d. 12 ft
20. which of the following statements is
ot\ true about similar figures?
a. side lengths are proportional.
b. a dilation can create similar figures.
c. corresponding angles are congruent.
d. similar figures must be the same size.
21. a rectangle with vertices p(0, 0), q(4, 0), r(4, 3), and s(0, 3) is translated to p(2, -1), q(6, -1), r(6, 2), and s(2, 2). what is the translation rule?
a. {x + 2, y + 1}
b. {x + 2, y - 1}
c. {x + 1, y + 2}
d. {x - 2, y - 1}

Question

15. what is the length of the hypotenuse of a right triangle with legs measuring 7 units and 24 units?
 a. 25 units
 b. 26 units
 c. 27 units
 d. 25 units
16. if triangle abc is similar to triangle def and the length of ab is 5 units while the length of de is 10 units, what is the scale factor from triangle abc to triangle def?
 a. 10
 b.
 c. 2
 d. 4
17. if a triangle is divided into two smaller triangles by an altitude and all three triangles are similar, what theorem could this be used to prove?
 a. sss criterion
 b. pythagorean theorem
 c. hl criterion
 d. sas criterion
18. rectangle r with side lengths 4 and 6 is dilated by a scale factor of 2. what are the side lengths of the resulting rectangle?
 a. 2 and 3
 b. 6 and 9
 c. 8 and 12
 d. 10 and 15
19. a ladder leaning against a building creates a right triangle with the ground. the ladder is 20 ft long, and the base of the ladder is 16 ft from the building. how high does the ladder reach on the building?
 a. 10 ft
 b. 15 ft
 c. 11 ft
 d. 12 ft
20. which of the following statements is
ot\ true about similar figures?
 a. side lengths are proportional.
 b. a dilation can create similar figures.
 c. corresponding angles are congruent.
 d. similar figures must be the same size.
21. a rectangle with vertices p(0, 0), q(4, 0), r(4, 3), and s(0, 3) is translated to p(2, -1), q(6, -1), r(6, 2), and s(2, 2). what is the translation rule?
 a. {x + 2, y + 1}
 b. {x + 2, y - 1}
 c. {x + 1, y + 2}
 d. {x - 2, y - 1}

Accepted Answer

# Explanation:
## Step1: Recall Pythagorean theorem
For a right - triangle with legs $a = 7$ and $b = 24$, the Pythagorean theorem is $c=\sqrt{a^{2}+b^{2}}$, where $c$ is the hypotenuse.
## Step2: Calculate $a^{2}+b^{2}$
$a^{2}=7^{2}=49$, $b^{2}=24^{2}=576$. Then $a^{2}+b^{2}=49 + 576=625$.
## Step3: Find the square root
$c=\sqrt{625}=25$.

# Answer:
d. 25 units

# Explanation:
## Step1: Define scale factor
The scale factor from $\triangle ABC$ to $\triangle DEF$ is $\frac{DE}{AB}$. Given $AB = 5$ and $DE = 10$.
## Step2: Calculate the scale factor
$\text{Scale factor}=\frac{DE}{AB}=\frac{10}{5}=2$.

# Answer:
c. 2

# Explanation:
## Step1: Recall the Pythagorean theorem proof
When a right - triangle is divided into two smaller similar right - triangles by an altitude, the Pythagorean theorem can be proved using the similarity of these triangles.

# Answer:
b. Pythagorean Theorem

# Explanation:
## Step1: Apply the scale factor formula
If a rectangle with side lengths $l$ and $w$ is divided by a scale factor $k$, the new side lengths are $\frac{l}{k}$ and $\frac{w}{k}$. Here $l = 4$, $w = 6$, $k = 2$.
## Step2: Calculate new side lengths
$\frac{4}{2}=2$ and $\frac{6}{2}=3$.

# Answer:
a. 2 and 3

# Explanation:
## Step1: Use the Pythagorean theorem
Let the height on the building be $h$. The ladder length $c = 20$ (hypotenuse), base $a = 16$. By $c^{2}=a^{2}+h^{2}$, we get $h=\sqrt{c^{2}-a^{2}}$.
## Step2: Calculate $c^{2}-a^{2}$
$c^{2}=20^{2}=400$, $a^{2}=16^{2}=256$. Then $c^{2}-a^{2}=400 - 256 = 144$.
## Step3: Find the square root
$h=\sqrt{144}=12$.

# Answer:
d. 12 ft

# Explanation:
## Step1: Analyze similar figures properties
Similar figures have proportional side lengths (a is true), congruent corresponding angles (c is true), and a dilation can create similar figures (b is true). But similar figures do not have to be the same size (d is false).

# Answer:
d. Similar figures must be the same size

# Explanation:
## Step1: Analyze the translation of $x$ - coordinates
For $P(0,0)$ to $P'(2,-1)$, $x$ - coordinate changes from $0$ to $2$ ($0+2$), $y$ - coordinate changes from $0$ to $-1$ ($0 - 1$). For $Q(4,0)$ to $Q'(6,-1)$, $x$ - coordinate $4+2$, $y$ - coordinate $0 - 1$.

# Answer:
b. $\{x + 2,y-1\}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Recall Pythagorean theorem

Step2: Calculate \(a^{2}+b^{2}\)

Step3: Find the square root

Step1: Define scale factor

Step2: Calculate the scale factor

Step1: Recall the Pythagorean theorem proof

Step1: Apply the scale factor formula

Step2: Calculate new side lengths

Answer: