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.
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Step 1: Solve Question 4. The problem states that triangles ABC and BXY are similar ( ABC BXY). We need to calculate the enlargement factor. From the diagram, we can identify corresponding sides. The triangles are oriented such that B is the center of enlargement. Side AC corresponds to side XY, and side AB corresponds to side XB. Given lengths: AC = 8 cm XY = 20 cm The enlargement factor (k) is the ratio of a side length in the enlarged triangle to the corresponding side length in the original triangle. Since XY (20 cm) is longer than AC (8 cm), BXY is the enlarged triangle and ABC is the original. k = Length of side in enlarged triangleLength of corresponding side in original triangle k = (XY)/(AC) k = 20 cm8 cm Simplify the fraction: k = (20 ÷ 4)/(8 ÷ 4) k = (5)/(2) The enlargement factor is (5)/(2). Step 2: Solve Question 5. The problem states that triangles ABC and XYZ are similar ( ABC XYZ). We need to find the length of side BC. Given lengths: In ABC: AB = 2 cm, BC = x In XYZ: XY = 5.6 cm, YZ = 14 cm, XZ = 11.2 cm Since the triangles are similar, the ratio of their corresponding sides is equal. The corresponding sides are: AB corresponds to XY BC corresponds to YZ AC corresponds to XZ We can set up a proportion using the known side lengths and the unknown side x: (AB)/(XY) = (BC)/(YZ) Substitute the given values: 2 cm5.6 cm = (x)/(14 cm) To solve for x, multiply both sides by 14: x = (2)/(5.6) × 14 x = (28)/(5.6) x = 5 The length of BC is 5 cm. Step 3: Solve Question 6. We need to show that ABD = ACD in the given figure. Consider ABD and ACD. 1. Side AB is equal to side AC (given by the single tick marks). 2. Side BD is equal to side CD (given by the double tick marks). 3. Side AD is common to both triangles. Since all three corresponding sides are equal (AB=AC, BD=CD, AD=AD), the triangles ABD and ACD are congruent by the Side-Side-Side (SSS) congruence criterion. ABD ACD (SSS) Because the triangles are congruent, their corresponding angles are equal. Therefore, ABD corresponds to ACD, and thus: ABD = ACD This shows that the angle ABD is equal to the angle ACD. Step 4: Solve Question 7. The problem states that triangles DEF and PQF are similar ( DEF PQF). We are given the area of PQF and need to calculate the area of DEF. Given lengths: EF = 15 cm FQ = 10 cm Area of PQF = 25 cm^2 When two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides. From the similarity statement DEF PQF, side EF corresponds to side QF. The ratio of corresponding sides (k) is: k = (EF)/(QF) k = 15 cm10 cm k = (3)/(2) The ratio of the areas is k^2: Area( DEF)Area( PQF) = ((EF)/(QF))^2 Area( DEF)25 cm^2 = ((3)/(2))^2 Area( DEF)25 cm^2 = (9)/(4) Now, solve for Area( DEF): Area( DEF) = (9)/(4) × 25 cm^2 Area( DEF) = (225)/(4) cm^2 Area( DEF) = 56.25 cm^2 The area of triangle DEF is 56.25 cm^2. Last free one today — make it count tomorrow, or type /upgrade for unlimited.