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Similarity

Find scale factors between similar figures and use similarity to find unknown sides and angles.

Worked examples

Finding the scale factor between similar triangles

Straightforward

Problem

Triangle ABCABC is similar to triangle DEFDEF. AB=4AB = 4 cm and DE=6DE = 6 cm. Find the scale factor from △ABC\triangle ABC to △DEF\triangle DEF.

Finding a missing side using similarity

Moderate

Problem

Triangle ABC∼ABC \sim triangle DEFDEF. AB=8AB = 8 cm, DE=12DE = 12 cm, and BC=10BC = 10 cm. Find EFEF.

Finding a missing side in nested similar triangles

Challenging

Problem

In triangle PQRPQR, point SS lies on PQPQ and point TT lies on PRPR such that ST∥QRST \parallel QR. PS=6PS = 6 cm, SQ=9SQ = 9 cm, and QR=20QR = 20 cm. Find STST.

Practise

Q1·Straightforward
Triangle ABCABC is similar to triangle DEFDEF. AB=5AB = 5 cm and DE=10DE = 10 cm. Find the scale factor from △ABC\triangle ABC to △DEF\triangle DEF.
Q2·Straightforward
Triangle GHIGHI is similar to triangle JKLJKL. GH=6GH = 6 cm and JK=9JK = 9 cm. Find the scale factor from △GHI\triangle GHI to △JKL\triangle JKL.
Q3·Straightforward
Two similar pentagons are drawn. A side in the first pentagon measures 88 cm, and the corresponding side in the second pentagon measures 66 cm. Find the scale factor from the first pentagon to the second.
Q4·Straightforward
Rectangle ABCDABCD is similar to rectangle EFGHEFGH. AB=4AB = 4 cm and EF=10EF = 10 cm. Find the scale factor from ABCDABCD to EFGHEFGH.
Q5·Moderate
Triangle ABC∼ABC \sim triangle DEFDEF. AB=6AB = 6 cm, DE=9DE = 9 cm, and BC=10BC = 10 cm. Find EFEF, in cm.
Q6·Moderate
Triangle PQR∼PQR \sim triangle STUSTU. PQ=8PQ = 8 cm, ST=6ST = 6 cm, and QR=12QR = 12 cm. Find TUTU, in cm.
Q7·Moderate
Triangle ABC∼ABC \sim triangle DEFDEF. AB=5AB = 5 cm, BC=8BC = 8 cm, and DE=3DE = 3 cm. Find EFEF, in cm.
Q8·Moderate
Rectangle PQRS∼PQRS \sim rectangle WXYZWXYZ. PQ=10PQ = 10 cm, WX=15WX = 15 cm, and QR=6QR = 6 cm. Find XYXY, in cm.
Q9·Challenging
In triangle ABCABC, point DD lies on ABAB and point EE lies on ACAC such that DE∥BCDE \parallel BC. AD=4AD = 4 cm, AB=10AB = 10 cm, and BC=15BC = 15 cm. Find DEDE, in cm.
Q10·Challenging
Two straight lines intersect at point XX. Points AA and BB lie on one line, with XX between AA and BB. Points CC and DD lie on the other line, with XX between CC and DD. AC∥BDAC \parallel BD, AX=5AX = 5 cm, BX=15BX = 15 cm, and AC=7AC = 7 cm. Find BDBD, in cm.
Q11·Challenging
Triangle ABC∼ABC \sim triangle DEFDEF. ∠A=63°\angle A = 63° and ∠C=47°\angle C = 47°. Find ∠E\angle E, in degrees.
Q12·Challenging
Triangle ABC∼ABC \sim triangle DEFDEF. AB=7AB = 7 cm, BC=10BC = 10 cm, AC=13AC = 13 cm, and DE=14DE = 14 cm. Find the perimeter of △DEF\triangle DEF, in cm.