The value of [tex]\text{x}[/tex] is [tex]\boxed{x = 5}.[/tex]
Further Explanation:
There are many similarity rules and are as follows.
1. Angle Angle (AA)
2. Side SideSide (SSS)
3. Side Angle Side (SAS)
If two triangles are similar to each other than the ratios of the corresponding sides are equal.
Explanation:
The similar triangles are those in which all the corresponding angles are equal and the sides are proportional.
The [tex]\Delta {\text{ACB}}[/tex] and [tex]\Delta {\text{ACE}}[/tex] are similar to each other. Therefore, the ratios of the corresponding sides are equal.
[tex]\begin{aligned}\frac{{{\text{AC}}}}{{{\text{DC}}}} &= \frac{{{\text{EC}}}}{{{\text{BC}}}} \\\frac{{{\text{AD}} + {\text{DC}}}}{{{\text{DC}}}} &= \frac{{{\text{EC}}}}{{{\text{BC}}}} \\\frac{{12 + 3}}{{12}} &= \frac{{4x}}{{3x + 1}}\\\frac{{15}}{{12}} &= \frac{{4x}}{{3x + 1}}\\\end{aligned}[/tex]
Further solve the above equation.
[tex]\begin{aligned}\frac{5}{4}&=\frac{{4x}}{{3x + 1}}\\5\left( {3x + 1} \right) &= 4\left( {4x} \right)\\15x + 5 &= 16x\\5&= 16x - 15x\\5&= x\\\end{aligned}[/tex]
The value of [tex]\text{x}[/tex] is [tex]\boxed{x = 5}.[/tex]
Kindly refer to the image attached.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Triangle
Keywords: congruent, angles, triangle, similar, value, x, two right triangles, one smaller, right angles, straight angle, two acute angles, overlapping, legs, labeled, 20, extends, smaller triangle, larger triangle, ray, 3x.
