(a) In The Figure (i) Given Below, ABCD Is A Trapezium In Which DC Is Parallel To AB. If AB = 9 Cm, DC 6 Cm And BD = 12 Cm, Find = (i) BP (ii) The Ratio Of Areas Of Triangle APB And Triangle DPC. (b) In The Figure (ii) Given Below, Angle ABC = Angle

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Introduction

In this problem, we are given a trapezium ABCD in which DC is parallel to AB. We are also given the lengths of AB, DC, and BD. Our goal is to find the length of BP and the ratio of areas of triangle APB and triangle DPC.

Given Information

  • AB = 9 cm
  • DC = 6 cm
  • BD = 12 cm

Solution

To solve this problem, we can use the properties of similar triangles. Since DC is parallel to AB, we can draw a line from point A to point D, which intersects BD at point P.

(i) Finding the length of BP

We can use the concept of similar triangles to find the length of BP. Since triangle APB is similar to triangle DPC, we can set up a proportion:

\frac{AP}{DP} = \frac{AB}{DC}

We know that AB = 9 cm and DC = 6 cm. We can substitute these values into the proportion:

\frac{AP}{DP} = \frac{9}{6}

We can simplify the proportion by dividing both sides by 3:

\frac{AP}{DP} = \frac{3}{2}

Now, we can use the fact that AP + DP = BD to find the length of BP. We know that BD = 12 cm, so we can set up the equation:

AP + DP = 12

We can substitute the expression for AP in terms of DP into the equation:

\frac{3}{2}DP + DP = 12

We can simplify the equation by combining like terms:

\frac{5}{2}DP = 12

We can solve for DP by multiplying both sides by 2/5:

DP = \frac{24}{5}

Now, we can find the length of BP by subtracting DP from BD:

BP = BD - DP

We can substitute the values of BD and DP into the equation:

BP = 12 - \frac{24}{5}

We can simplify the equation by finding a common denominator:

BP = \frac{60}{5} - \frac{24}{5}

We can combine the fractions:

BP = \frac{36}{5}

Therefore, the length of BP is 36/5 cm.

(ii) Finding the ratio of areas of triangle APB and triangle DPC

We can use the concept of similar triangles to find the ratio of areas of triangle APB and triangle DPC. Since triangle APB is similar to triangle DPC, we can set up a proportion:

\frac{[APB]}{[DPC]} = \frac{AB^2}{DC^2}

We know that AB = 9 cm and DC = 6 cm. We can substitute these values into the proportion:

\frac{[APB]}{[DPC]} = \frac{9^2}{6^2}

We can simplify the proportion by evaluating the squares:

\frac{[APB]}{[DPC]} = \frac{81}{36}

We can simplify the proportion by dividing both sides by 9:

\frac{[APB]}{[DPC]} = \frac{9}{4}

Therefore, the ratio of areas of triangle APB and triangle DPC is 9/4.

Conclusion

In this problem, we used the properties of similar triangles to find the length of BP and the ratio of areas of triangle APB and triangle DPC. We can use these concepts to solve a wide range of problems involving similar triangles.

Introduction

In this problem, we are given a triangle ABC in which angle ABC = angle DCA. We are also given the lengths of AB and AC. Our goal is to find the length of BC.

Given Information

  • AB = 6 cm
  • AC = 8 cm

Solution

To solve this problem, we can use the concept of similar triangles. Since angle ABC = angle DCA, we can draw a line from point C to point B, which intersects AB at point E.

Finding the length of BC

We can use the concept of similar triangles to find the length of BC. Since triangle ABC is similar to triangle DCA, we can set up a proportion:

\frac{AB}{AC} = \frac{AE}{EC}

We know that AB = 6 cm and AC = 8 cm. We can substitute these values into the proportion:

\frac{6}{8} = \frac{AE}{EC}

We can simplify the proportion by dividing both sides by 2:

\frac{3}{4} = \frac{AE}{EC}

Now, we can use the fact that AE + EC = AC to find the length of BC. We know that AC = 8 cm, so we can set up the equation:

AE + EC = 8

We can substitute the expression for AE in terms of EC into the equation:

\frac{3}{4}EC + EC = 8

We can simplify the equation by combining like terms:

\frac{7}{4}EC = 8

We can solve for EC by multiplying both sides by 4/7:

EC = \frac{32}{7}

Now, we can find the length of BC by adding AE and EC:

BC = AE + EC

We can substitute the values of AE and EC into the equation:

BC = \frac{3}{4}EC + EC

We can substitute the value of EC into the equation:

BC = \frac{3}{4}\left(\frac{32}{7}\right) + \frac{32}{7}

We can simplify the equation by finding a common denominator:

BC = \frac{24}{7} + \frac{32}{7}

We can combine the fractions:

BC = \frac{56}{7}

Therefore, the length of BC is 56/7 cm.

Conclusion

In this problem, we used the concept of similar triangles to find the length of BC. We can use these concepts to solve a wide range of problems involving similar triangles.

In this article, we solved two problems involving similar triangles. In the first problem, we used the properties of similar triangles to find the length of BP and the ratio of areas of triangle APB and triangle DPC. In the second problem, we used the concept of similar triangles to find the length of BC. We can use these concepts to solve a wide range of problems involving similar triangles.

References

Further Reading

Final Thoughts

In conclusion, similar triangles are a powerful tool for solving problems in geometry. By using the properties of similar triangles, we can find the lengths of sides, the areas of triangles, and much more. We hope that this article has provided a helpful introduction to the concept of similar triangles and has inspired you to learn more about this fascinating topic.

Introduction

Similar triangles are a fundamental concept in geometry that can be used to solve a wide range of problems. In this article, we will answer some of the most frequently asked questions about similar triangles.

Q: What are similar triangles?

A: Similar triangles are two or more triangles that have the same shape, but not necessarily the same size. This means that the corresponding angles of the triangles are equal, and the corresponding sides are proportional.

Q: How do I know if two triangles are similar?

A: To determine if two triangles are similar, you can use the following criteria:

  • The corresponding angles of the triangles are equal.
  • The corresponding sides of the triangles are proportional.

Q: What is the ratio of similarity?

A: The ratio of similarity is a measure of how similar two triangles are. It is calculated by dividing the length of a side of one triangle by the length of the corresponding side of the other triangle.

Q: How do I find the ratio of similarity?

A: To find the ratio of similarity, you can use the following formula:

\frac{a}{b} = \frac{c}{d}

where a and b are the lengths of the corresponding sides of the two triangles, and c and d are the lengths of the corresponding sides of the other two triangles.

Q: What is the relationship between the areas of similar triangles?

A: The areas of similar triangles are proportional to the square of the ratio of similarity. This means that if the ratio of similarity is 2:1, the area of the larger triangle is 4 times the area of the smaller triangle.

Q: How do I find the area of a similar triangle?

A: To find the area of a similar triangle, you can use the following formula:

\frac{[A]}{[B]} = \left(\frac{a}{b}\right)^2

where [A] and [B] are the areas of the two triangles, and a and b are the lengths of the corresponding sides.

Q: What is the relationship between the perimeters of similar triangles?

A: The perimeters of similar triangles are proportional to the ratio of similarity. This means that if the ratio of similarity is 2:1, the perimeter of the larger triangle is 2 times the perimeter of the smaller triangle.

Q: How do I find the perimeter of a similar triangle?

A: To find the perimeter of a similar triangle, you can use the following formula:

\frac{P_A}{P_B} = \frac{a}{b}

where P_A and P_B are the perimeters of the two triangles, and a and b are the lengths of the corresponding sides.

Q: Can similar triangles be used to solve real-world problems?

A: Yes, similar triangles can be used to solve a wide range of real-world problems. For example, they can be used to calculate the height of a building, the distance between two points, or the area of a room.

Q: What are some common applications of similar triangles?

A: Some common applications of similar triangles include:

  • Calculating the height of a building or a tree
  • Determining the distance between two points
  • Finding the area of a room or a piece of land
  • Calculating the volume of a container or a solid object
  • Determining the length of a shadow or a projection

Conclusion

In conclusion, similar triangles are a powerful tool for solving problems in geometry. By understanding the properties and relationships of similar triangles, you can solve a wide range of problems and apply them to real-world situations.

References

Further Reading

Final Thoughts

In conclusion, similar triangles are a fundamental concept in geometry that can be used to solve a wide range of problems. By understanding the properties and relationships of similar triangles, you can apply them to real-world situations and solve problems in a variety of fields.