Question 965195: PQRS is a rectangle in which PQ = 9cm and PS = 6cm. T is a point on PQ such that PT = 7cm and RV is the perpendicular from R to ST. Find RV. (1dp)
Thanks!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the area of triangle STR is equal to 1/2 times the area of rectangle PQRS.
if you know this property of triangles that have their base equal to one side of a rectangle and their vertex a point on the opposite side of the rectangle, then the rest becomes easy.
the area of rectangle PQRS is equal to 9 * 6 = 54 square cm.
the area of triangle STR is therefore equal to 1/2 of 54 which is equal to 27 square cm.
the area of a triangle is equal to 1/2 * the base * the height.
that would make the area of triangle STR equal to 1/2 * ST * VR, because ST is the base of triangle STR and VR is the height of triangle STR when the altitude of the triangle is drawn from point R to point V on ST.
ST is also the hypotenuse of right triangle SPT.
that makes ST equal to sqrt (6^2 + 7^2) = sqrt(85).
you know the area of triangle STR is equal to 27 square cm.
you know the base of triangle STR is equal to ST which is equal to sqrt(85).
you can use the area formula to find the height of triangle STR which is equal to VR.
area = 1/2 * base * height becomes:
27 = 1/2 * sqrt(85) * VR.
solve for VR to get VR = 54 / sqrt(85).
here's a picture of what i just talked about.
that's your solution.
why is the area of a triangle formed within a rectangle equal to 1/2 * the area of the rectangle.
the area of the rectangle is equal to length * width.
the length is the base of the rectangle and the width is the height.
the triangle formed with its base being the same as the length of the rectangle and its height being the same as the height of the rectangle will have an area of 1/2 * length of the rectangle * width of the rectangle which is equal to 1/2 * the area of the rectangle.
you can see this relationship in the following diagram:
it doesn't matter where point T is.
the area of triangle STR is always equal to 1/2 * 6 * 9 = 27 regardless of where point T is.
note that any side of a triangle can be equal to the base of the triangle.
whether that side is a base or not is determined by where the altitude of the triangle is drawn to.
in triangle STR, if the altitude is drawn from point T to point U on SR, then the base of triangle STR is SR (see the second diagram).
in the same triangle STR, the altitude is drawn from point R to point V on ST, then the base of triangle STR is ST (see the first diagram).
same triangle, but different bases because different altitude were drawn from different vertices of the same triangle the sides opposite those vertices.
to solve this problem, you needed to draw TR because that showed you what the area of triangle STR was.
if you didn't know that the area of triangle STR was 1/2 times the area of rectangle PQRS, you would have had to find the area of STR anyway.
that would have been by finding the area of rectangle PQRS and triangle SPT and triangle TQR and then subtracting the the area of those triangles from the area of the rectangle to get the area of triangle STR.
this is not a lot more difficult, but it does require more calculations.
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