Question

Work out the unkown lenghts image provided

Answer 1

`x= (36)/(5)`

`y=(15)/(4)`

Explanation:

Triangles formed by transversals of parallel lines

When two parallel lines are intersected by a transversal, the resulting alternate interior angles are congruent. By introducing a second transversal that intersects the initial one, two similar triangles are created. This happens because the angles formed at the point where the two transversals intersect are considered vertically opposite angles.

In similar triangles:

• Corresponding angles have the same measure.
• Corresponding sides are always in the same ratio.

The corresponding sides of the given triangles are as follows:

• Side lengths: 3 mm and y
• Side lengths: 4 mm and 5 mm
• Side lengths: x and 9 mm

To find the values of x and y, we can set up the following ratio:

`(3)/(y)=(4)/(5)=(x)/(9)`

Solve for x:

`\begin{aligned}(4)/(5)&=(x)/(9)\\\\(4)/(5)\cdot 9&=(x)/(9)\cdot 9\\\\x&=(36)/(5)\end{aligned}`

Solve for y:

`\begin{aligned}(3)/(y)&=(4)/(5)\\\\3 \cdot 5&=4 \cdot y\\\\15&=4y\\\\y&=(15)/(4)\end{aligned}`

Therefore, the values of x and y are:

`\large\boxed{\boxed{x= (36)/(5)\quad y=(15)/(4)}}`

`\hrulefill`

Additional Notes

The side lengths given on the diagram do not form valid triangles.

According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

For the smaller triangle, 3 + 4 > x.

`\textsf{As $x = (36)/(5) = 7.2$, then $3 + 4 \\ot > 7.2$}`

Therefore, we cannot construct a triangle with side lengths 3, 4 and 7.2.

For the larger triangle, 5 + y > 9.

`\textsf{As $y = (15)/(4) = 3.75$, then $5+3.75 \\ot > 9$}`

Therefore, we cannot construct a triangle with side lengths 3.75, 5 and 9.