Parallelogram Solutions

L2MAEL

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Question One:

Equation for the long sides of the box can be found by using the equation formula:

$$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$$

The bottom long side of the box has an equation of:

$$y=\frac{1}{4}x$$

and the top long side of the box has an equation of:

$$y=\frac{1}{4}x+5$$

Since:

$$\underset{\text{Gradient 1}}{\underbrace{\frac{1}{4}}}=\underset{\text{Gradient 2}}{\underbrace{\frac{1}{4}}}$$

The lines are parallel.

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Question Two:

Equation of the black triangle can be found by using:

$$Area=\frac{1}{2}bh$$

Base is the distance between the origin and $(4,1)$. Using the distance formula:

$$Dist=\sqrt{(y_2-y_1{)}^2+(x_2-x_1{)}^2}$$

The base is $\sqrt{5}$.

The height can be be found by drawing a line that’s perpendicular to the baseline, and making it pass through the point $(1,5.25)$. f The baseline has a gradient of $\frac{1}{4}$, which indicates that the perpendicular gradient of the baseline would be $-4$, since $\frac{1}{4}\times -4=-1U$.

Using equation formula:

$$y-5.25=-4(x-1)$$ $$y=-4x+4+5.25$$ $$y=-4x+9.25$$

Now, find the intersection of the perpendicular line and the baseline by using systems:

$$\{\begin{array}{r}y=\frac{1}{4}x\\ y=-4x+9.25\end{array}$$ $$\frac{1}{4}x=-4x+9.25$$ $$4.25x=9.25$$ $$x=2.1765\text{ (4dp)}$$

Substitute back into baseline:

$$y=\frac{1}{4}(2.1765)$$ $$y=0.5441$$

Hence, the coordinate of the perpendicular line that intersects with the baseline is:

$$(2.1765,0.5441)$$

To find the height, now use distance formula between the coordinate that I just found and the top corner of the triangle.

$$Height=\sqrt{(5.25-0.54410{)}^2+(1-2.1765{)}^2}$$ $$Height=4.8507$$

Now, substitute everything back into the area formula:

$$Area=\frac{1}{2}\times \sqrt{5}\times 4.8507$$ $$Area=10{\text{units}}^2\text{ (4dp)}$$

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Question Three & Four:

While there are many ways that this can be found, the black, red, and blue triangles have the same base and height, and thus the same area.

This question involves ‘critical thinking’ as one of its solutions, and this response counts for both question three and four.

While there should be a way to calculate the coordinate of the top corner of the red and blue triangles, it may be too complicated for Level 2. Hence, using critical thinking may be easier for people when solving this question.

The ‘critical thinking’ part is using the nature of parallel lines. If a parallel line is given, any and all perpendicular lines formed between the parallel lines should have the same distance as other perpendicular lines formed within the parallel lines. If not, then the two long sides wouldn’t be parallel.

Since the red and blue triangles have the same base as the black triangle, the perpendicular gradient (height) of the red and blue triangles would be the same. Hence, the red and blue triangles would have the same height as the black triangle.

Therefore, the red and blue triangles have the same area as they have the same height and the base.