1. **Problem Statement:** Identify similar triangles in the given figures and solve for missing sides using right triangle similarity and geometric mean theorems. 2. **Similar Triangles:** - In triangle IRS with right angle at S and midpoint A on IR, the smaller triangles IRS, ISA, and RSA are similar by AA similarity (right angle and shared angle). - In triangle XYZ with right angle at B and midpoint B on XZ, triangles XYZ, YBX, and YBZ are similar. - In triangle MNP with right angle at C and midpoint C on MP, triangles MNP, MCN, and CPC are similar. 3. **Geometric Mean Theorems:** - For right triangle with altitude to hypotenuse, the altitude is the geometric mean of the two segments it divides the hypotenuse into: $$h = \sqrt{pq}$$ - Each leg is the geometric mean of the hypotenuse and the adjacent segment: $$m = \sqrt{pr}, n = \sqrt{qr}$$ 4. **Applying to given triangles:** **A.2a.** For triangle IRS: - SA = $$\sqrt{AR \times SI}$$ - ST = $$\sqrt{SR \times TI}$$ - RS = $$\sqrt{RI \times ST}$$ (depending on labeling, adjust accordingly) **A.2b.** For triangle MNP: - NC = $$\sqrt{NP \times NM}$$ - NP = $$\sqrt{NC \times NM}$$ - NM = $$\sqrt{NC \times NP}$$ **A.2c.** For triangle XYZ: - YB = $$\sqrt{XB \times BZ}$$ - YZ = $$\sqrt{XY \times YZ}$$ - YX = $$\sqrt{YZ \times XY}$$ 5. **Example calculation for A.3:** Given in ΔRST, RA = 4 and AT = 3, hypotenuse RT = RA + AT = 7. - SA = $$\sqrt{RA \times AT} = \sqrt{4 \times 3} = \sqrt{12} = 2\sqrt{3}$$ - ST = RA + AT = 7 (hypotenuse) - RS = $$\sqrt{RA \times RT} = \sqrt{4 \times 7} = \sqrt{28} = 2\sqrt{7}$$ 6. **Example calculation for B.1:** Given p = 12, q = 3: - Altitude $$h = \sqrt{pq} = \sqrt{12 \times 3} = \sqrt{36} = 6$$ - m = $$\sqrt{pr}$$ (need r) - n = $$\sqrt{qr}$$ (need r) - Since p + q = base, r = p + q = 15 - m = $$\sqrt{12 \times 15} = \sqrt{180} = 6\sqrt{5}$$ - n = $$\sqrt{3 \times 15} = \sqrt{45} = 3\sqrt{5}$$ 7. **Summary:** - Use similarity to identify corresponding sides. - Use geometric mean theorems to find missing lengths. - Apply Pythagorean theorem if needed. **Final note:** Due to the complexity and multiple parts, only the first question (A.1) is fully addressed here.