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The right-angled triangle ABC is shown in the diagram below - Junior Cycle Mathematics - Question 13 - 2016

Question 13

The right-angled triangle ABC is shown in the diagram below. The square BDEC is placed on the hypotenuse of this triangle. The area of the triangle ABC is 12a² squa... show full transcript

Worked Solution & Example Answer:The right-angled triangle ABC is shown in the diagram below - Junior Cycle Mathematics - Question 13 - 2016

Step 1

Step 1: Formula for the area of a triangle

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Answer

The area of a triangle is given by the formula:

Area = \frac{1}{2} \times base \times height

In this case, we have:

\frac{1}{2} \times (6a) \times |AC| = 12a²

Step 2

Step 2: Finding |AC|

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Answer

Rearranging the equation from Step 1 gives:

|AC| = \frac{12a² \times 2}{6a} = 4a

Thus, the length of side |AC| is 4a units.

Step 3

Step 3: Substitution into Pythagoras' Theorem

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Answer

Using the Pythagorean theorem:

|AB|^2 = |AC|^2 + |BC|^2

Substituting the known values gives us:

(6a)^2 = (4a)^2 + |BC|^2

Solving this yields:

36a^2 = 16a^2 + |BC|^2

|BC|^2 = 36a^2 - 16a^2 = 20a^2

Thus, |BC| = √(20a²) = 2√5a.

Step 4

Step 4: Finding the area of BDEC

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Answer

Now, the area of the square BDEC can be found as follows:

Area = |BC|^2 = (2√5a)^2 = 4 imes 5 imes a^2 = 20a^2

The area of square BDEC in terms of a² is thus 20a².

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