Part 1: ABCD is a square and O is the intersection of the diagonals. (a) Is triangle ABO and triangle ADO congruent? (b) What is the measure of angle AOB? If OA = 6 cm, then what i... Part 1: ABCD is a square and O is the intersection of the diagonals. (a) Is triangle ABO and triangle ADO congruent? (b) What is the measure of angle AOB? If OA = 6 cm, then what is the value of CA? (c) If OB = 6 cm, then what is OD? Part 2: Anuj, Manish, Deepak, and Shubham went for a picnic and made a conical tent. They had 200 m^2 of cloth. The tent has a height of 8m and a diameter of 12m. The remaining cloth was used for the floor. (π = 3.14)

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Understand the Problem

The question involves two parts: the first part consists of geometry-related questions regarding a square with diagonals, asking about triangle congruence, angle measures, and side lengths. The second part is a word problem about calculating the amount of cloth used to make a conical tent and its floor, given the total available cloth.

Answer

(a) Yes (b) $90^\circ$ (c) $12$ cm (d) $6$ cm
Answer for screen readers

(a) Yes, triangle $ABO$ and triangle $ADO$ are congruent. (b) The measure of angle $AOB$ is $90^\circ$. (c) If $OA = 6$ cm, then the value of $CA$ is $12$ cm. (d) If $OB = 6$ cm, then $OD = 6$ cm.

Steps to Solve

  1. Congruence of triangles ABO and ADO

    We need to determine if triangles $ABO$ and $ADO$ are congruent in square $ABCD$. $AB = AD$ because all sides of a square are equal. $AO = AO$ is a common side. $\angle BAO = \angle DAO = 45^\circ$ because the diagonals of a square bisect the angles. Therefore, $\triangle ABO \cong \triangle ADO$ by the Side-Angle-Side (SAS) congruence criterion.

  2. Measure of angle AOB

    The diagonals of a square bisect each other at right angles. Therefore, $\angle AOB = 90^\circ$.

  3. Value of CA if OA = 6 cm

    $O$ is the intersection of the diagonals, so $O$ is the midpoint of $CA$.

    Since $OA = 6$ cm, then $CA = 2 \times OA = 2 \times 6 = 12$ cm.

  4. Value of OD if OB = 6 cm

    In a square, the diagonals are equal and bisect each other. Thus, if $OB = 6$ cm, then $OD$ must also be 6 cm.

(a) Yes, triangle $ABO$ and triangle $ADO$ are congruent. (b) The measure of angle $AOB$ is $90^\circ$. (c) If $OA = 6$ cm, then the value of $CA$ is $12$ cm. (d) If $OB = 6$ cm, then $OD = 6$ cm.

More Information

The diagonals of a square have the following properties:

  • They are equal in length.
  • They bisect each other at right angles.
  • They bisect the angles at the vertices. These properties are key to solving the geometric questions posed.

Tips

A common mistake is assuming that triangle $ABO$ and triangle $ADO$ are not congruent without proper justification. Another common mistake is forgetting that the diagonals of a square bisect each other at right angles. In the third question, students may forget that point $O$ bisects line $CA$, and $CA = 2 * OA$. null

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