Nalanda College, Colombo 10 Grade 11 Maths Worksheet: 12 Equiangular Triangles

Unknown segment from a line parallel to a side

Answer: x = 4 cm

💡 What it's asking: DE is parallel to BC, so it cuts AB and AC in the same ratio. Write that ratio as an equation and solve for x.

1. 1A line drawn parallel to one side of a triangle cuts the other two sides in the same ratio. So AD : DB = AE : EC.
2. 2Put in the lengths: (10 − x) : x = 3 : 2.
3. 3Cross-multiply: 2(10 − x) = 3x, so 20 − 2x = 3x.
4. 4Add 2x to both sides: 20 = 5x, so x = 4 cm.

✅ With x = 4, AD = 6 and DB = 4, and 6 : 4 = 3 : 2, the same as AE : EC.

Solve for x using parallel-line ratio

Answer: x = 2

💡 What it's asking: ST is parallel to QR, so the two sides from P are cut in the same ratio. Set up the ratio and solve for x.

1. 1A line drawn parallel to one side of a triangle cuts the other two sides in the same ratio. So PS : SQ = PT : TR.
2. 2Put in the lengths: (x − 1) : 3 = 2 : 6.
3. 3Simplify 2 : 6 to 1 : 3, so (x − 1) : 3 = 1 : 3.
4. 4The second terms are both 3, so x − 1 = 1 and x = 2.

✅ With x = 2, PS = 1 and 1 : 3 = 2 : 6, so the ratios match.