(04.03 MC) Calculate the area of triangle ABC with altitude CD, given A (6,0), B (1,5), C (2,0), and D (4,2). O 5 square units

Question

(04.03 MC)
Calculate the area of triangle ABC with altitude CD, given A (6,0), B (1,5), C (2,0), and D (4,2).
O 5 square units
O 8 square units
O 10 square units
O 13 square units
Question 7

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RI SƠ 5 years 2021-07-21T04:07:49+00:00 1 Answers 451 views 2

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  1. Giakhanh
    1
    2021-07-21T04:09:38+00:00
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    Answer:

    10 square units.

    Step-by-step explanation:

    There are three vertices in this triangle: \rm A, \rm B, and \rm C. The three sides are \rm AB, \rm BC, and \rm AC.

    Among the two endpoints of altitude \rm CD, only \rm C is a vertex of this triangle. Hence, \rm AB, the side opposite to vertex \rm C\!, would be the base of this altitude.

    Apply the Pythagorean Theorem to find the length of \rm AB (the base) and \rm CD (the height).

    By the Pythagorean Theorem, the distance between points (x_0,\, y_0) and (x_1,\, y_1) is \sqrt{(x_1 - x_0)^{2} + (y_1 - y_0)^{2}}.

    The distance between \rm C (2,\, 0) and \rm D (4,\, 2) is:

    \sqrt{(4 - 2)^{2} + (2 - 0)^{2}} = \sqrt{8} = 2\, \sqrt{2}.

    Hence, the length of altitude \rm CD would be 2\sqrt{2} units.

    Similarly, the length of side \rm AB would be:

    \sqrt{(6 - 1)^{2} + (0 - 5)^{2}} = \sqrt{50} = 5\, \sqrt{2}.

    Calculate the area of this triangle:

    \begin{aligned}& \text{area of triangle} \\ &= \frac{1}{2} \times (\text{base}) \times(\text{height}) \\ &= \frac{1}{2} \times (\text{length of AB}) \times (\text{length of CD})\\ &= \frac{1}{2} \times 5\sqrt{2} \times 2\sqrt{2} = 10\end{aligned}.

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