Can you find area of the Blue shaded region? | (Nice Geometry problem) | #math #maths | #geometry

PreMath
17 Jun 202409:51

TLDRIn this geometry problem, a blue shaded quadrilateral is inscribed in a right triangle with a 45° angle and a 90° angle, given segment lengths of AE as 4 units and EB as 2 units. Using the Triangle Sum Theorem and the Triangle Proportionality Theorem, the video demonstrates how to find the area of the quadrilateral by calculating the area ratios of similar triangles and concludes that the area of the blue region is 12 square units. The video is an engaging exploration of geometric principles applied to a complex figure.

Takeaways

  • 📐 The blue shaded quadrilateral BFD is fully confined in a right triangle ABC with specific angle measures.
  • 🔍 Angle BAC is 45°, and angle ADE is 90°, with segment AE being 4 units and EB being 2 units.
  • 📏 Segment BF is 5 units, and FC is 1 unit, which are key measurements for the problem.
  • 📚 The Triangle Sum Theorem is used to determine the angles within the triangles of the problem.
  • 📏 By dropping a perpendicular BP, the Triangle Proportionality Theorem is applied to find the ratio of segments on side AC.
  • 🔄 Triangle AD is similar to triangle AB by the Angle-Angle Similarity Theorem, leading to a proportional ratio of segments.
  • 🔢 The ratio of segments BE to AE is simplified to 1/2, helping to determine the lengths of other segments in the problem.
  • 📐 Triangle ABC is identified as an isosceles triangle with BP as the perpendicular bisector, which gives equal lengths for AP and PC.
  • 📏 The lengths of segments DC and AD are calculated, leading to a ratio of 2:4 for AD to CD.
  • 📐 The area ratio of triangles with the same height is the same as the ratio of their bases, which is applied to find the area of the blue shaded region.
  • 🔢 The area of triangle ABC is calculated to be 18 square units using the formula for the area of a triangle.
  • 📝 The final calculation concludes that the area of the blue shaded region is 12 square units.

Q & A

  • What is the main objective of the video?

    -The main objective of the video is to calculate the area of the blue shaded quadrilateral BFD confined within a right triangle ABC.

  • What are the given angles and side lengths in the problem?

    -The given angles are ∠BAC = 45°, ∠ADE = 90°. The given side lengths are AE = 4 units, EB = 2 units, BF = 5 units, and FC = 1 unit.

  • What theorem is used to deduce the angles in the right triangles?

    -The Triangle Sum Theorem is used to deduce that the other angles in the right triangles are also 45°.

  • How does the video establish that triangle AD is similar to triangle AB?

    -By using the Angle-Angle Similarity Theorem, which states that if two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.

  • What is the Triangle Proportionality Theorem, and how is it applied in the video?

    -The Triangle Proportionality Theorem states that if a line parallel to one side of a triangle intersects the other two sides, it divides the sides proportionally. In the video, it is applied to find the ratio of segments on side AC.

  • How does the video determine the lengths of segments DP and AD?

    -By setting up a proportion based on the Triangle Proportionality Theorem and knowing the lengths of BE and AE, the video deduces that if DP is 1, then AD is 2, making the whole segment AC 3 units.

  • What observation is made about triangle ABC that helps in the calculation?

    -The observation that triangle ABC is isosceles with BP as the perpendicular bisector is made, which implies that AP = PC.

  • How is the ratio of the segments AD and CD determined?

    -By using the fact that AP = PC and knowing the lengths of AD and DC, the ratio of AD to CD is determined to be 2:4, which simplifies to 1:2.

  • What principle is used to relate the areas of triangles with the same height?

    -The principle that triangles with the same height have areas whose ratio is the same as the ratio of their bases is used.

  • How does the video calculate the area of triangle ABC?

    -The video uses the formula for the area of a triangle, which is 1/2 * base * height, with the base and height both being 6 units, resulting in an area of 18 square units.

  • What is the final calculated area of the blue shaded region?

    -The final calculated area of the blue shaded region is 12 square units.

Outlines

00:00

📐 Introduction to the Geometry Problem

This paragraph introduces a geometry problem involving a blue shaded quadrilateral within a right triangle. The right triangle ABC has a 45° angle at BAC and a 90° angle at AD. Given lengths for segments AE, EB, BF, and FC are 4 units, 2 units, 5 units, and 1 unit respectively. The task is to calculate the area of the blue region. The video emphasizes the importance of the Triangle Sum Theorem and the properties of similar triangles and parallel lines in solving the problem. The script also notes that the diagram may not be perfectly to scale.

05:00

🔍 Applying Triangle Proportionality Theorem and Similarity

The second paragraph delves into the application of the Triangle Proportionality Theorem, which states that if a line parallel to one side of a triangle intersects the other two sides, it divides them proportionally. This is used to establish the similarity between triangle AD and triangle AB, leading to the calculation of segment lengths and ratios. The paragraph also discusses the properties of an isosceles triangle with a perpendicular bisector, leading to the determination of segment lengths on the triangle's sides. The focus then shifts to the area ratios of triangles with the same height, using the base lengths to find the relationship between their areas.

Mindmap

Keywords

💡Quadrilateral

A quadrilateral is a four-sided polygon with four vertices. In the context of the video, the blue shaded region is a quadrilateral that is fully confined within a right triangle. The understanding of the properties of a quadrilateral is essential to solving the problem as it helps in determining the area of the blue shaded region.

💡Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. In the video, the right triangle ABC is the larger geometric figure within which the blue shaded quadrilateral is situated. The properties of right triangles are used to calculate angles and side lengths relevant to finding the area of the shaded region.

💡Angle

An angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex. The video mentions specific angles such as 45° at BAC and 90° at ADF, which are crucial for understanding the geometric relationships and for applying the Triangle Sum Theorem.

💡Triangle Sum Theorem

The Triangle Sum Theorem states that the sum of the interior angles of a triangle is always 180 degrees. The video uses this theorem to deduce the measures of unknown angles within the triangles ABC and ADE, which is key to further geometric analysis.

💡Segment

A segment in geometry refers to a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. The video script provides specific segment lengths such as AE being 4 units and EB being 2 units, which are necessary for calculating ratios and areas within the geometric figures.

💡Proportionality Theorem

The Proportionality Theorem, also known as Thales' theorem, states that if a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally. In the video, this theorem is used to establish the ratio of segments on side AC, which is essential for finding the lengths of segments within the triangles.

💡Similar Triangles

Similar triangles are triangles that have the same shape but may differ in size, with their corresponding angles being equal and their corresponding sides being in proportion. The video identifies that triangle AD is similar to triangle AB, which allows for the calculation of unknown lengths based on known proportions.

💡Area Ratio

The area ratio of two shapes is the ratio of their areas. In the context of the video, the area ratio is used to compare the areas of different triangles within the figure, which is crucial for determining the total area of the blue shaded region.

💡Base and Height

In a triangle, the base is any side chosen as the reference from which the height is measured, and the height is the perpendicular distance from the base to the opposite vertex. The video uses the base and height of triangles to calculate their areas using the formula: area = 1/2 * base * height.

💡Area Calculation

Area calculation is the process of determining the size of a two-dimensional shape. In the video, the area of the right triangle ABC and the blue shaded quadrilateral is calculated using geometric formulas and the properties of the shapes involved. This is the ultimate goal of the problem presented in the video.

Highlights

The problem involves finding the area of a blue shaded quadrilateral within a right triangle with specific angles and side lengths.

Angle BAC is 45°, and angle ADE is 90°, with AE being 4 units and EB being 2 units.

BF is 5 units, and FC is 1 unit, which are key measurements for the problem.

The Triangle Sum Theorem is used to determine the angles within the triangles.

Triangle proportionality theorem is applied to find the ratio of segments on side AC.

A perpendicular line BP is drawn parallel to ED to aid in the proportionality analysis.

Triangles AD and AB are similar by the angle-angle similarity theorem, leading to a proportion of their sides.

The ratio of DP to AD is found to be equal to the ratio of BE to AE, simplifying to 1/2.

Observation that triangle ABC is isosceles with BP as the perpendicular bisector.

AP and PC lengths are determined to be equal, both being 3 units.

DC length is calculated as 4 units, and AD length as 2 units, establishing a new ratio.

The area ratio of triangles with the same height is the same as the ratio of their bases.

Triangles ABD and BCD are analyzed for their area ratios based on their bases.

The area of the entire triangle ABC is calculated to be 9 using the area ratios.

The numeric area of triangle ABC is calculated using the formula: area = 1/2 * base * height.

The area of triangle ABC is found to be 18 square units.

Equating the two methods of finding the area of triangle ABC leads to a contradiction that is resolved.

The area of the blue shaded region is determined to be 12 square units by scaling the area ratio.

The video concludes with the final area calculation and a prompt to subscribe for more content.

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