Triangle Geometry: Angle Bisector Theorem
The problem involves finding the length of the third side of a triangle using the Angle Bisector Theorem.
Given:
- Triangle $\Delta\text{XYZ}$
- Side lengths: $\text{XY} = 12\text{ cm}$, $\text{YZ} = 18\text{ cm}$
- XW is the angle bisector of $\angle\text{YXZ}$, meeting $\text{YZ}$ at W.
- Ratio: $\text{YW} : \text{WZ} = 4 : 5$
- To find: Length of side $\text{XZ}$.
Applying the Angle Bisector Theorem
The Angle Bisector Theorem states that an angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle.
For $\Delta\text{XYZ}$ and angle bisector XW:
$ \frac{\text{YW}}{\text{WZ}} = \frac{\text{XY}}{\text{XZ}} $Calculation
We are given the ratio $\frac{\text{YW}}{\text{WZ}} = \frac{4}{5}$ and the side length $\text{XY} = 12\text{ cm}$.
Substitute these values into the Angle Bisector Theorem equation:
$ \frac{4}{5} = \frac{12\text{ cm}}{\text{XZ}} $To find $\text{XZ}$, rearrange the equation:
$ \text{XZ} = \frac{12\text{ cm} \times 5}{4} $Calculate the result:
$ \text{XZ} = \frac{60\text{ cm}}{4} $ $ \text{XZ} = 15\text{ cm} $Result
The length of the third side, $\text{XZ}$, is $15\text{ cm}$.
Similar Questions
- In $\triangle ABC$, $BD \perp AC$ at $D$ and $\angle DBC = 40^\circ$. $E$ is a point on $BC$ such that $\angle CAE = 37^\circ$. What is the measure of $\angle AEB$?
- In $\Delta ABC$, $BD \perp AC$ at D and $\angle DBC = 71^\circ$. E is a point on BC such that $\angle CAE = 17^\circ$. What is the measure of $\angle AEB$?
- In triangle ABC, bisector of $\angle\text{ABC}$ and $\angle\text{ACB}$ meet at O. If $\angle\text{BAC} = 60^\circ$, then find the measure of $\angle\text{BOC}$.
- In $\Delta\text{PQR}$, $\text{QR}$ is extended up to $\text{S}$ so that $\text{RS} = \text{RP}$. If $\angle\text{PRQ} = 70^\circ$ and $\angle\text{QPS} = 110^\circ$ then find the measure of $\angle\text{PQS}$.
- The areas of two similar triangles are respectively $16\text{ m}^{2}$ and $36\text{ m}^{2}$. Find the ratio of their corresponding sides.
- In $\Delta PQR$, QR is extended up to S, so that RS = RP. If $\angle RPQ = 55^\circ$ and $\angle PRS = 110^\circ$, then find the measure of $\angle PQS$.
- The sides of a triangle are 44 cm, 33 cm, and 55 cm. What is its area? (in $cm^2$)
- The perimeter of an equilateral triangle ABC is 22.2 cm. What is the area of triangle (in $cm^2$)?
- In $\triangle\text{ABC}$, $\text{BD} \perp \text{AC}$ at $\text{D}$ and $\angle\text{DBC} = 60^{\circ}$. $\text{E}$ is a point on $\text{BC}$ such that $\angle\text{CAE} = 20^{\circ}$. What is the measure of $\angle\text{AEB}$?
- In a triangle, if angle $A = 30^\circ$ and angle $B = 45^\circ$, then what is the angle of C?
Important Questions from Triangles
What is the circumcenter of the triangle ABC?
What is the centroid of the triangle ABC?
What is the foot of the altitude from the vertex A of the triangle ABC?
In ΔABC, D is a point on BC such that ∠ADB = 2∠DAC, ∠BAC = 70° and ∠B = 56°. What is the measure of ∠ADC?
In ΔABC, ∠A = 66° and ∠B = 50 °. If the bisectors of ∠B and ∠C meet at P, then ∠BPC – ∠PCA = ?
