Name: Transforming Algebraic Functions
Uploaded: 2019-07-19
Duration: 21 min 54 s
Description: We know the basics of graphing algebraic functions; it's time to learn some tricks that will come in handy as we graph different functions.

Question

A rectangle is inscribed in isosceles triangle $A B C$ as shown in the diagram.The altitude of triangle $A B C$ from $B$ to side $A C$ is $7$ cm and $A C = 8$ cm. The coordinates of one of the vertices of the inscribed rectangle are $(p, 0)$ . Find the dimensions of the rectangle inscribed in the triangle, in terms of $p$ .

Accepted Answer

Given that, a rectangle is inscribed in a triangle $A B C$

A attitude is drawn from a vertex to the side $A C$ i.e., $B O$ .

Therefore, $A C = 8$ cm and $(p, 0)$ cm

So, clearly, the co-ordinates of the point $B (0, 7)$

Similarly, the length of side $A C = 8$ cm. Therefore, $O$ is the mid-point of side $A C$

Thus, the co-ordinates of the points $A (- 4, 0)$ and $C (4, 0)$ .

Hence, the required co-ordinates of the points: $A (- 4, 0), B (0, 7) and C (4, 0)$ .

Now, the equation of the line passing through points $B (0, 7) and C (4, 0)$ is $y - 7 = \frac{0 - 7}{4 - 0} (x - 0)$

$\Rightarrow 4 y - 28 = - 7 x$

$\Rightarrow 4 y = - 7 x + 28$

$\Rightarrow y = - 1.75 x + 7$ , which is the required equation of a line.

Let coordinates of the points of rectangle are $P, Q, R, S$ .

Therefore, the point on $x -$ axis is $P (- p, 0)$

From , figure, the coordinates of the point $R (p, y)$ and similarly, the coordinates of the point $S (- p, y)$

It is clear, that, the point $R (p, y)$ lie on the side $B C$ and its equation of line is $y = - 1.75 x + 7$

Therefore, $y = - 1.75 p + 7$

Thus, the coordinates of points $S (- p, - 1.75 p + 7)$ and $R (p, - 1.75 p + 7)$

So, the dimensions of the rectangle inscribed in triangle $A B C$ are: length $= P Q$ and breadth $= Q R$

Therefore, the length $P Q = 2 p$ cm and breadth $Q R = - 1.75 p + 7$ cm.

Important Questions on Modelling Relationships: Linear and Quadratic Functions

Learn and Prepare for any exam you want !

A rectangle is inscribed in isosceles triangle ABC as shown in the diagram.The altitude of triangle ABC from B to side AC is 7cm and AC=8 cm. The coordinates of one of the vertices of the inscribed rectangle are p,0. Find the dimensions of the rectangle inscribed in the triangle, in terms of p.

Important Questions on Modelling Relationships: Linear and Quadratic Functions

Learn and Prepare for any exam you want !