phase 2: medium levels (questions 7-14)

focus: calculating unknown sides, checking for right triangles, and standard word problems.

7. a right triangle has a leg of $5\text{ cm}$ and a hypotenuse of $13\text{ cm}$. calculate the length of the missing leg.

8. a construction worker measures the three sides of a wooden frame to see if the corner forms a perfect $90^\circ$ angle. the side measurements are $6\text{ cm}$, $8\text{ cm}$, and $11\text{ cm}$. is this frame a true right triangle? (answer yes or no, and show your work).

9. a $10\text{-meter}$ long ladder is leaning against the side of a house. if the base of the ladder is placed exactly $6\text{ meters}$ away from the wall, how high up the side of the house will the ladder reach?

10. find the length of the hypotenuse $c$ for a right triangle with legs $a = 8\text{ cm}$ and $b = 15\text{ cm}$.

11. a square piece of land has an area of $150\text{ m}^2$. estimate the side length of this land to one decimal place.

12. a laptop screen has a width of $12\text{ inches}$ and a height of $9\text{ inches}$. laptop sizes are advertised by the diagonal measure of the screen. what size is this laptop?

13. look at the three squares attached to the sides of the right triangle below. the area of square a is $25\text{ cm}^2$ and the area of square c (the hypotenuse square) is $169\text{ cm}^2$. what is the area of square b?

14. a rectangle measures $7\text{ cm}$ wide by $24\text{ cm}$ long. calculate the exact length of the diagonal line cutting through the center of this rectangle.

Question

phase 2: medium levels (questions 7-14)

focus: calculating unknown sides, checking for right triangles, and standard word problems.

7. a right triangle has a leg of $5\text{ cm}$ and a hypotenuse of $13\text{ cm}$. calculate the length of the missing leg.

8. a construction worker measures the three sides of a wooden frame to see if the corner forms a perfect $90^\circ$ angle. the side measurements are $6\text{ cm}$, $8\text{ cm}$, and $11\text{ cm}$. is this frame a true right triangle? (answer yes or no, and show your work).

9. a $10\text{-meter}$ long ladder is leaning against the side of a house. if the base of the ladder is placed exactly $6\text{ meters}$ away from the wall, how high up the side of the house will the ladder reach?

10. find the length of the hypotenuse $c$ for a right triangle with legs $a = 8\text{ cm}$ and $b = 15\text{ cm}$.

11. a square piece of land has an area of $150\text{ m}^2$. estimate the side length of this land to one decimal place.

12. a laptop screen has a width of $12\text{ inches}$ and a height of $9\text{ inches}$. laptop sizes are advertised by the diagonal measure of the screen. what size is this laptop?

13. look at the three squares attached to the sides of the right triangle below. the area of square a is $25\text{ cm}^2$ and the area of square c (the hypotenuse square) is $169\text{ cm}^2$. what is the area of square b?

14. a rectangle measures $7\text{ cm}$ wide by $24\text{ cm}$ long. calculate the exact length of the diagonal line cutting through the center of this rectangle.

Accepted Answer

{ "quality": "clear", "question_count": 8, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Pythagorean Theorem", "Square Roots" ], "new_concepts": [], "current_concepts": [ "Pythagorean Theorem", "Square Roots", "Right Triangles" ] } ## Calculate the missing leg for Question 7 $$ \begin{aligned} a^2 + b^2 &= c^2 \ 5^2 + b^2 &= 13^2 \ 25 + b^2 &= 169 \ b^2 &= 144 \ b &= 12\text{ cm} \end{aligned} $$ ## Verify the right triangle for Question 8 $$ \begin{aligned} a^2 + b^2 &= c^2 \ 6^2 + 8^2 &= 36 + 64 = 100 \ 11^2 &= 121 \ 100 & eq 121 \implies \text{No} \end{aligned} $$ ## Calculate the height of the ladder for Question 9 $$ \begin{aligned} a^2 + b^2 &= c^2 \ 6^2 + h^2 &= 10^2 \ 36 + h^2 &= 100 \ h^2 &= 64 \ h &= 8\text{ m} \end{aligned} $$ ## Calculate the hypotenuse for Question 10 $$ \begin{aligned} c^2 &= a^2 + b^2 \ c^2 &= 8^2 + 15^2 \ c^2 &= 64 + 225 = 289 \ c &= 17\text{ cm} \end{aligned} $$ ## Estimate the side length of the square for Question 11 $$ \begin{aligned} s^2 &= 150 \ s &= \sqrt{150} \approx 12.2\text{ m} \end{aligned} $$ ## Calculate the diagonal laptop size for Question 12 $$ \begin{aligned} d^2 &= 12^2 + 9^2 \ d^2 &= 144 + 81 = 225 \ d &= 15\text{ inches} \end{aligned} $$ ## Calculate the area of Square B for Question 13 $$ \begin{aligned} \text{Area}_A + \text{Area}_B &= \text{Area}_C \ 25 + \text{Area}_B &= 169 \ \text{Area}_B &= 144\text{ cm}^2 \end{aligned} $$ ## Calculate the diagonal of the rectangle for Question 14 $$ \begin{aligned} d^2 &= 7^2 + 24^2 \ d^2 &= 49 + 576 = 625 \ d &= 25\text{ cm} \end{aligned} $$ | No. | Answer | |---|---| | 7 | $12\text{ cm}$ | | 8 | No, $6^2 + 8^2 = 100 eq 11^2$ | | 9 | $8\text{ m}$ | | 10 | $17\text{ cm}$ | | 11 | $12.2\text{ m}$ | | 12 | $15\text{ inches}$ | | 13 | $144\text{ cm}^2$ | | 14 | $25\text{ cm}$ | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Pythagorean Theorem" ] }

9	\(8\text{ m}\)
10	\(17\text{ cm}\)
11	\(12.2\text{ m}\)
12	\(15\text{ inches}\)
13	\(144\text{ cm}^2\)
14	\(25\text{ cm}\)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Calculate the missing leg for Question 7

Verify the right triangle for Question 8

Calculate the height of the ladder for Question 9

Calculate the hypotenuse for Question 10

Estimate the side length of the square for Question 11

Calculate the diagonal laptop size for Question 12

Calculate the area of Square B for Question 13

Calculate the diagonal of the rectangle for Question 14

Answer: