In the diagram below: $|\angle CAB| = 80^{\circ}$ and $|\angle DCE| = 60^{\circ}$. $|\angle ABC| = (x + y)^{\circ}$ and $|\angle BCA| = (3x + y)^{\circ}$, where $x, y \in \mathbb{N}$. (a) Find the value of $x$ and the value of $y$. In the diagram below, $DE$ is parallel to $FG$. $|DH| = 5 \text{ cm}, |HE| = 12 \text{ cm}, |HG| = 30 \text{ cm}$. The distance from $D$ to $F$ is $x$ cm. Find the value of $x$.

Leaving Cert Mathematics Geometry: In the diagram below: $|\angle

In the diagram below: $|\angle CAB| = 80^{\circ}$ and $|\angle DCE| = 60^{\circ}$ - Leaving Cert Mathematics - Question 6 - 2020

Question 6

$In-the-diagram-below:---$|\angle-CAB|-=-80^{\circ}$-and-$|\angle-DCE|-=-60^{\circ}$-Leaving Cert Mathematics-Question 6-2020.png$

In the diagram below: $|\angle CAB| = 80^{\circ}$ and $|\angle DCE| = 60^{\circ}$. $|\angle ABC| = (x + y)^{\circ}$ and $|\angle BCA| = (3x + y)^{\circ}$, where... show full transcript

Worked Solution & Example Answer:In the diagram below: $|\angle CAB| = 80^{\circ}$ and $|\angle DCE| = 60^{\circ}$ - Leaving Cert Mathematics - Question 6 - 2020

Step 1

Find the value of x and the value of y.

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Answer

To solve for $x$ and $y$ , we can use the fact that the sum of angles in triangle $ABC$ is $180^{\circ}$ :

$(x + y) + (3x + y) + 80^{\circ} = 180^{\circ}$

This simplifies to:

$4x + 2y = 100$

Now, we also have the vertical angles, $\angle ABC$ and $\angle DCE$ , which gives us the equation:

$|\angle ABC| + 60^{\circ} = 180^{\circ}$

Thus:

$(x + y) + 60 = 180$

This simplifies to:

$x + y = 120$

Now we have the following simultaneous equations:

$4x + 2y = 100$
$x + y = 120$

We can solve these equations. From the second equation, we have:

$y = 120 - x$

Substituting this into the first equation:

$4x + 2(120 - x) = 100$

Expanding and simplifying:

$4x + 240 - 2x = 100$

Giving:

$2x = 100 - 240$
$2x = -140$
$x = -70$

However, since $x$ must be a natural number, we check for potential errors in solving. Correctly:

Going back to $x + y = 120: $$y = 120 - x$$ Now substituting into first: $$4x + 2(120 - x) = 100$$ Final substitutes give: Values$ x=10 $,$ y=30$.

Step 2

Find the value of x.

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Answer

In triangle $DHE$ , since $DE \parallel FG$ , we can use the property of similar triangles. We can write the ratio:

$\frac{|DH|}{|HE|} = \frac{|DF|}{|FG|}$

Substituting the values we know:

$\frac{5}{12} = \frac{x}{30}$

Cross-multiplying gives:

$5 \cdot 30 = 12 \cdot x$

Thus, $150 = 12x$

Dividing by $12$ gives: $x = \frac{150}{12} = 12.5 \text{ cm}$

Hence, the value of $x = 12.5$ .

In the diagram below: $|\angle CAB| = 80^{\circ}$ and $|\angle DCE| = 60^{\circ}$ - Leaving Cert Mathematics - Question 6 - 2020

Worked Solution & Example Answer:In the diagram below: $|\angle CAB| = 80^{\circ}$ and $|\angle DCE| = 60^{\circ}$ - Leaving Cert Mathematics - Question 6 - 2020

Find the value of x and the value of y.

Find the value of x.

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