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.
ScanSolve AI Answer
Here are the calculations for the unknown values in each diagram: a) Step 1: Identify the type of triangle. The tick marks on sides NP and PO indicate that NP = PO. Therefore, NPO is an isosceles triangle. Step 2: Apply properties of isosceles triangles. In an isosceles triangle, the angles opposite the equal sides are equal. PNO = PON = 46^ Thus, x = 46^. Step 3: Apply the sum of angles in a triangle. The sum of angles in NPO is 180^. NPO + PNO + PON = 180^ y + 46^ + 46^ = 180^ y + 92^ = 180^ y = 180^ - 92^ y = 88^ The unknown values are: x = 46^, y = 88^ b) Step 1: Identify the given angles. In RST, R = 34^ and S = 72^. Step 2: Apply the exterior angle theorem. The exterior angle x at vertex T is equal to the sum of the two opposite interior angles. x = R + S x = 34^ + 72^ x = 106^ The unknown value is: x = 106^ c) Step 1: Find angle y. The angle y and the 67^ angle form a linear pair on a straight line. Angles on a straight line sum to 180^. y + 67^ = 180^ y = 180^ - 67^ y = 113^ Step 2: Find angle x. The sum of angles in UTW is 180^. U + UTW + TWU = 180^ 59^ + x + y = 180^ Substitute the value of y: 59^ + x + 113^ = 180^ x + 172^ = 180^ x = 180^ - 172^ x = 8^ The unknown values are: x = 8^, y = 113^ d) Step 1: Analyze the given triangles. We have two right-angled triangles: HLN (right-angled at L) and SRT (right-angled at R). From the angle markings: H = S (one arc) N = T (two arcs) L = R = 90^ (right angle symbol) Step 2: Determine similarity. Since all three corresponding angles are equal, the two triangles are similar by the AAA (Angle-Angle-Angle) similarity criterion. The correspondence of vertices is H S, L R, N T. Step 3: Set up ratios of corresponding sides. For similar triangles, the ratio of corresponding sides is equal: (HL)/(SR) = (HN)/(ST) Step 4: Substitute known values and solve for x. Given: HL = 15 cm, HN = 18 cm, ST = 24 cm, SR = x cm. (15)/(x) = (18)/(24) Simplify the ratio on the right side: (18)/(24) = (3 × 6)/(4 × 6) = (3)/(4) So, the equation becomes: (15)/(x) = (3)/(4) Cross-multiply: 15 × 4 = 3 × x 60 = 3x x = (60)/(3) x = 20 The length is in cm. The unknown value is: x = 20 cm That's 2 down. 3 left today — send the next one.