If Δ ABC~Δ FDE such that AB = 9 cm, AC = 11 cm, DF = 16 cm and DE = 12 cm, then the length of BC is:
Understanding Similar Triangles and Proportional Sides
When two triangles are similar, it means their corresponding angles are equal, and their corresponding sides are in proportion. The symbol '∼' denotes similarity. If $\Delta$ ABC $\sim$ $\Delta$ FDE, it means:
- ∠A = ∠F
- ∠B = ∠D
- ∠C = ∠E
And the ratio of their corresponding sides is constant:
$$ \frac{AB}{FD} = \frac{BC}{DE} = \frac{AC}{FE} $$In this problem, we are given that $\Delta$ ABC $\sim$ $\Delta$ FDE. We are provided with the lengths of certain sides of these similar triangles.
Given Information for Similar Triangles ABC and FDE
We are given the following side lengths:
- AB = 9 cm
- AC = 11 cm
- DF = 16 cm
- DE = 12 cm
We need to find the length of side BC.
Finding Corresponding Sides in Similar Triangles
The similarity statement $\Delta$ ABC $\sim$ $\Delta$ FDE tells us which vertices correspond:
- A corresponds to F
- B corresponds to D
- C corresponds to E
Based on this correspondence, the pairs of corresponding sides are:
- AB corresponds to FD
- BC corresponds to DE
- AC corresponds to FE
Setting up the Proportion to Calculate BC
Since the triangles are similar, the ratio of corresponding sides is equal. We can use the sides we know to find the ratio, and then use that ratio to find the unknown side BC. The proportional relationship is:
$$ \frac{AB}{FD} = \frac{BC}{DE} = \frac{AC}{FE} $$We have the lengths for AB, FD, and DE, and we need to find BC. So, we can use the proportion involving these sides:
$$ \frac{AB}{FD} = \frac{BC}{DE} $$Step-by-Step Calculation of BC Length
Now, substitute the given values into the proportion:
$$ \frac{9 \text{ cm}}{16 \text{ cm}} = \frac{BC}{12 \text{ cm}} $$To solve for BC, we can multiply both sides of the equation by 12 cm:
$$ BC = \frac{9}{16} \times 12 \text{ cm} $$Perform the multiplication:
$$ BC = \frac{9 \times 12}{16} \text{ cm} $$ $$ BC = \frac{108}{16} \text{ cm} $$Now, simplify the fraction $\frac{108}{16}$. Both 108 and 16 are divisible by 4.
$$ BC = \frac{108 \div 4}{16 \div 4} \text{ cm} $$ $$ BC = \frac{27}{4} \text{ cm} $$The options are given in mixed number form. Let's convert the improper fraction $\frac{27}{4}$ into a mixed number. Divide 27 by 4:
27 divided by 4 is 6 with a remainder of 3. So, $\frac{27}{4}$ can be written as $6 \frac{3}{4}$.
$$ BC = 6 \frac{3}{4} \text{ cm} $$Thus, the length of side BC is $6 \frac{3}{4}$ cm.
Revision Table: Similar Triangles Summary
| Concept | Description | Application in Problem |
|---|---|---|
| Similar Triangles | Triangles with equal corresponding angles and proportional corresponding sides. | Δ ABC ∼ Δ FDE is given. |
| Corresponding Sides | Sides opposite corresponding angles in similar triangles. Their ratios are equal. | AB and FD, BC and DE, AC and FE are corresponding pairs. |
| Proportion | An equation stating that two ratios are equal. Used to find unknown side lengths. | Used $\frac{AB}{FD} = \frac{BC}{DE}$ to solve for BC. |
Additional Information: Properties of Similar Triangles
Beyond proportional sides, similar triangles have other important properties:
- Angles: Corresponding angles are congruent (equal).
- Perimeter Ratio: The ratio of the perimeters of two similar triangles is equal to the ratio of their corresponding sides.
- Area Ratio: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. If the side ratio is $k$, the area ratio is $k^2$.
- Altitude/Median/Angle Bisector Ratio: The ratio of corresponding altitudes, medians, or angle bisectors is also equal to the ratio of the corresponding sides.
Understanding these properties is crucial for solving various geometry problems involving similar triangles.
Similar Questions
Let ABC, PQR be two congruent triangles such that angle A = angle P = 90°. If BC = 13 cm, PR = 5 cm, find AB.
ΔABC ~ ΔDEF and the perimeters of ΔABC and ΔDEF are 40 cm and 12 cm, respectively. If DE = 6 cm, then AB is:
ΔABC ∼ ΔPQR, ar (ΔABC) = 16 cm2 and ar (ΔPQR) = 25 cm2. If BC = 20 cm, then QR is equal to :
In a ΔABC, DE ∥ BC, where D is a point on AB and E is a point on AC. If DE divides the area of ΔABC into two equal parts, then DB ∶ AB is equal to :
The centroid of an equilateral triangle PQR is L. If PQ = 6 cm, the length of PL is:
From the circumcentre L of ΔXYZ, perpendicular LM is drawn on side YZ. If ∠YXZ = 60°, then the measure of ∠YLM is :
In an equilateral triangle ABC, D is the midpoint of side BC. If the length of BC is 8 cm, then the height of the triangle is:
In a ΔABC, the median BE intersects AC at E. If BG = 12 cm, where G is the centroid, then BE is equal to:
ΔABC ∼ ΔDEF such that AB = 9.1 cm and DE = 6.5 cm. If the perimeter of ΔDEF = 25 cm, then the perimeter of ΔABC is:
If the angles of a triangle are in the ratio of 1 ∶ 2 ∶ 3, what is the type of such triangle?
Important Questions from Triangles, Congruence and Similarity
The radius of the circumcircle of an equilateral triangle of √3 unit side, is:
If the ratio of the angles of a triangle is 3 : 5 : 7, find the value of the largest angle.
A. 36°
B. 60°
C. 84°
D. 15°
If ΔABC and Δ PQR are similar and \(\rm\frac{BC}{QR} = \frac{1}{3}\) , find \(\rm\frac{ar(\Delta PQR)} {ar(\Delta BCA)}\)
If the angles of a triangle are in the ratio of 2 : 5 : 8, then find the value of the smallest angle.
ABCD is a parallelogram. Side BC is produced to E such that BC = CE. Join AE which intersects side CD at P. The area of triangle ABE is:
