1 The two triangles in the diagram are similar. A x cm B 2 cm E C 9 cm D 3 cm There are two possible values of x. Work out each of these values. State any assumptions you make in your working. (5 marks) Adapted from IMAI/IH, November 2017, Q22
The Correct Answer and Explanation is:
To solve this problem, we need to apply the concept of similar triangles. When two triangles are similar, their corresponding sides are proportional. From the information given, we know the following:
- Triangle ABC is similar to triangle CDE.
- The sides of triangle ABC are proportional to the sides of triangle CDE.
- We are given that side AB = x cm, BC = 2 cm, CD = 9 cm, and DE = 3 cm.
We can set up a proportion based on the similarity of the triangles. The corresponding sides of the triangles will be in the same ratio:ABCD=BCDE\frac{{AB}}{{CD}} = \frac{{BC}}{{DE}}CDAB=DEBC
Step 1: Write the proportion
Using the values provided, we can substitute into the proportion:x9=23\frac{{x}}{{9}} = \frac{{2}}{{3}}9x=32
Step 2: Solve for x
To solve for x, we can cross-multiply and solve:x×3=2×9x \times 3 = 2 \times 9x×3=2×93x=183x = 183x=18x=183=6 cmx = \frac{{18}}{{3}} = 6 \, \text{cm}x=318=6cm
This gives one possible value for x.
Step 3: Consider the reverse ratio
Since the triangles are similar, we also know that the sides can be related by the reverse proportion:CDAB=DEBC\frac{{CD}}{{AB}} = \frac{{DE}}{{BC}}ABCD=BCDE
Substitute the known values:9x=32\frac{{9}}{{x}} = \frac{{3}}{{2}}x9=23
Step 4: Solve for x again
Cross-multiply to find x:9×2=3×x9 \times 2 = 3 \times x9×2=3×x18=3×18 = 3×18=3xx=183=6 cmx = \frac{{18}}{{3}} = 6 \, \text{cm}x=318=6cm
Thus, the only possible value for x is 6 cm.
Conclusion
In this case, we find that x = 6 cm. The assumption we made was that the triangles are indeed similar and that their corresponding sides are proportional.
