In the figure triangle PQR, PQ = 8cm, PR = 12cm and S is a point on PR such that ∠PQS = ∠PRQ. ⓘ Write down the other pair of equal angles. ⓘ Use similar triangles to calculate the length of PS. ⓘ Given that the area of triangle PQS is 24cm², calculate the area of triangle PQR.

Question

Asked on July 23, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-07-23T05:34:00.352Z

Question 3: Step 1: Understand the relationship between the ratio of sides and the ratio of areas for similar triangles. If two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Let the ratio of corresponding sides be k = (s_1)/(s_2). The ratio of their areas is Area_1Area_2 = k^2 = ((s_1)/(s_2))^2. Step 2: Substitute the given ratio of sides and calculate the ratio of areas. Given the ratio of corresponding sides is 4:7, so k = (4)/(7). Area_1Area_2 = ((4)/(7))^2 = (4^2)/(7^2) = (16)/(49) The ratio of their areas is 16:49. Question 10: Given: In PQR, PQ = 8 cm, PR = 12 cm. S is a point on PR such that PQS = PRQ. The diagram also indicates that QS PR, meaning QSP = 90^. i) Write down the other pair of equal angles. Step 1: Identify common and given equal angles in PQS and PQR. We have: • QPS = QPR (Common angle) • PQS = PRQ (Given) Step 2: Conclude similarity and identify the third pair of equal angles. Since two pairs of corresponding angles are equal, PQS PQR by the Angle-Angle (AA) similarity criterion. Therefore, the third pair of corresponding angles must also be equal: QSP = PQR The other pair of equal angles is QSP = PQR. ii) Use similar triangles to calculate the length of PS. Step 1: Write down the ratio of corresponding sides from the similarity established in part (i). Since PQS PQR, the ratio of their corresponding sides is equal: (PS)/(PQ) = (PQ)/(PR) = (QS)/(QR) Step 2: Substitute the known values and solve for PS. We use the ratio (PS)/(PQ) = (PQ)/(PR). Given PQ = 8 cm and PR = 12 cm. (PS)/(8) = (8)/(12) PS = 8 × (8)/(12) PS = (64)/(12) PS = (16)/(3) cm The length of PS is (16)/(3) cm. iii) Given that the area of triangle PQS is 24 cm^2. Calculate the area of triangle PQR. Step 1: Determine the ratio of corresponding sides. From part (ii), the ratio of corresponding sides is (PQ)/(PR) = (8)/(12) = (2)/(3). Step 2: Use the relationship between the ratio of areas and the ratio of sides for similar triangles. The ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides. Area( PQS)Area( PQR) = ((PQ)/(PR))^2 Area( PQS)Area( PQR) = ((2)/(3))^2 = (4)/(9) Step 3: Substitute the given area and solve for the area of PQR. Given Area( PQS) = 24 cm^2. (24)/(Area)( PQR) = (4)/(9) Area( PQR) = 24 × (9)/(4) Area( PQR) = 6 × 9 Area( PQR) = 54 cm^2 The area of triangle PQR is 54 cm^2. Question 11: The phrasing of this question, "of 8 cm. A similar rectangle has an area of 36 cm^2 calculate its length," is ambiguous. It is unclear what the "8 cm" refers to (e.g., a side length of the first rectangle, or its area, or a ratio), and what "its length" refers to (e.g., the length of the first rectangle, or the second rectangle). Without further clarification or additional information about the first rectangle, this question cannot be solved accurately. Send me the next one 📸

Accepted Answer · 2026-07-23T05:34:00.352Z

Question 3: Step 1: Understand the relationship between the ratio of sides and the ratio of areas for similar triangles. If two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Let the ratio of corresponding sides be k = (s_1)/(s_2). The ratio of their areas is Area_1Area_2 = k^2 = ((s_1)/(s_2))^2. Step 2: Substitute the given ratio of sides and calculate the ratio of areas. Given the ratio of corresponding sides is 4:7, so k = (4)/(7). Area_1Area_2 = ((4)/(7))^2 = (4^2)/(7^2) = (16)/(49) The ratio of their areas is 16:49. Question 10: Given: In PQR, PQ = 8 cm, PR = 12 cm. S is a point on PR such that PQS = PRQ. The diagram also indicates that QS PR, meaning QSP = 90^. i) Write down the other pair of equal angles. Step 1: Identify common and given equal angles in PQS and PQR. We have: • QPS = QPR (Common angle) • PQS = PRQ (Given) Step 2: Conclude similarity and identify the third pair of equal angles. Since two pairs of corresponding angles are equal, PQS PQR by the Angle-Angle (AA) similarity criterion. Therefore, the third pair of corresponding angles must also be equal: QSP = PQR The other pair of equal angles is QSP = PQR. ii) Use similar triangles to calculate the length of PS. Step 1: Write down the ratio of corresponding sides from the similarity established in part (i). Since PQS PQR, the ratio of their corresponding sides is equal: (PS)/(PQ) = (PQ)/(PR) = (QS)/(QR) Step 2: Substitute the known values and solve for PS. We use the ratio (PS)/(PQ) = (PQ)/(PR). Given PQ = 8 cm and PR = 12 cm. (PS)/(8) = (8)/(12) PS = 8 × (8)/(12) PS = (64)/(12) PS = (16)/(3) cm The length of PS is (16)/(3) cm. iii) Given that the area of triangle PQS is 24 cm^2. Calculate the area of triangle PQR. Step 1: Determine the ratio of corresponding sides. From part (ii), the ratio of corresponding sides is (PQ)/(PR) = (8)/(12) = (2)/(3). Step 2: Use the relationship between the ratio of areas and the ratio of sides for similar triangles. The ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides. Area( PQS)Area( PQR) = ((PQ)/(PR))^2 Area( PQS)Area( PQR) = ((2)/(3))^2 = (4)/(9) Step 3: Substitute the given area and solve for the area of PQR. Given Area( PQS) = 24 cm^2. (24)/(Area)( PQR) = (4)/(9) Area( PQR) = 24 × (9)/(4) Area( PQR) = 6 × 9 Area( PQR) = 54 cm^2 The area of triangle PQR is 54 cm^2. Question 11: The phrasing of this question, "of 8 cm. A similar rectangle has an area of 36 cm^2 calculate its length," is ambiguous. It is unclear what the "8 cm" refers to (e.g., a side length of the first rectangle, or its area, or a ratio), and what "its length" refers to (e.g., the length of the first rectangle, or the second rectangle). Without further clarification or additional information about the first rectangle, this question cannot be solved accurately. Send me the next one 📸

In the figure triangle PQR, PQ = 8cm, PR = 12cm and S is a point on PR such that ∠PQS = ∠PRQ. ⓘ Write down the other pair of equal angles. ⓘ Use similar triangles to calculate the length of PS. ⓘ Given that the area of triangle PQS is 24cm², calculate the area of triangle PQR.

Need help with your own homework?

More Mathematics Questions

In the figure triangle PQR, PQ = 8cm, PR = 12cm and S is a point on PR such that ∠PQS = ∠PRQ. ⓘ Write down the other pair of equal angles. ⓘ Use similar triangles to calculate the length of PS. ⓘ Given that the area of triangle PQS is 24cm², calculate the area of triangle PQR.

Need help with your own homework?

More Mathematics Questions