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hello welcome to we learn daily in this
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video I'll show you how to find the area
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of a triangle inside a rectangle given
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that the areas of the other triangles
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are 6 20 and 21 square
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units as always to assess the difficulty
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pause the video and try to solve the
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problem on your own let's start by
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denoting the legs of the right triangles
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on one side of the rectangle As A and B
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this means the entire side of the
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rectangle will be equal to a + b now
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let's denote the legs on the other side
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of the rectangle as C and D therefore
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the entire side of the rectangle will be
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C + D next we'll express everything
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given using a b c and d we'll start by
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recording the area of each right
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triangle as half the product of its legs
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so we get that the area of the first
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triangle is equal to a * C + D / 2 and
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21 for the second triangle we get D * A
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+ B / 2 and this equals
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20 and for the third triangle we get B *
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C / 2 and this equals 6 we multiply both
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sides of each equation by two and for
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the first two equations we can expand
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the brackets to find the area of the
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triangle inside the rectangle we can
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calculate the area area of the entire
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rectangle and then subtract the known
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triangles the area of the rectangle is
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equal to the product of its two sides
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after expanding the brackets we have a *
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C + a * D + B * D + B * C now we can
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observe that a * C + a * D according to
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the conditions equals 42 we also know
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that a * D + B * D according to the
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conditions equals 40 and B * C = 12 to
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find the right side we are missing one a
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* D if something is missing we need to
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add it so let's add a * D to each part
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of the last equation in the left part we
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get the area of the rectangle plus a * D
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on the right side we get 42 + 40 + 12
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which equal 94 now we need to express
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the area of the rectangle using a * D
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let's multiply both sides of the
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previous equation by a * D then the area
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of the rectangle multipli a * D will
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equal a * D * a + b and a c plus D since
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we already know the products we
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substitute the known values into the
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equation as a result we obtain 40 * 42
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for Simplicity let's denote the area of
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the rectangle as X and express a * D
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from the first equation and substitute
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equation expanding the brackets and
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transferring everything to the right
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side we obtain a quadratic equation
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since the sum of these roots equals 94
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according to vieta's theorem X can be
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equal to 70 or X can be equal to the
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other possible root however X represents
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the area of the entire rectangle which
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already includes the three right
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triangles with a total area greater than
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24 thus there is only one possible value
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rectangle the area of the rectangle is
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70 square units so the area of the
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triangle we need to find will be the
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difference between 70 and the sum of the
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areas of the three other triangles let's
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write down the answer the area of the
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triangle is equal to 23 square
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solved if you understood the solution
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give it a like leave a comment and don't
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forget to subscribe to the channel
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thank you for watching I hope to see you
